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I want to express my sincere gratitude to my dissertation advisor, Dr. Elif Akgahl, for her guidance, insight and support during this research and study. I am also grateful to my coninittee niembers, Dr. Farid AitSahlia, Dr. P. Oscar Boykin and Dr. Siriphong L leph. ii" ol..IsI ch, for their constructive II_ar; Hun and coninents. I wish to extend my warmest thanks to my mentor, Dr. Shangyao Yan, for leading me to the field of operations research, his friendship and numerous fruitful discussions. My deepest gratitude goes to to my parents, ZheXiong and FangXue, and my husband, ChlungtJui. Without their understanding and encouragement, it would have been impossible for me to complete my degree. TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES. LIST OF FIGURES ABSTRACT CHAPTER 1 INTRODUCTION 2 AGGREGATE HOSPITAL BED CAPACITY PLANNING ....... 2.1 Introduction. 2.2 Literature Review. 2.3 Problem Formulation. 2.3.1 Restricted Bed Capacity Planning Problem. 2.3.2 Restricted Bed Capacity Planning Problem with Shu 2.4 Illustration of the Model. 2.4.1 A Representative DecisionMaking Scenario. 2.4.2 Experiment 1 An Application of the Model. 2.4.3 Experiment 2 Assessing the Impact of Problem Pa 2.5 Extensions. 2.6 Concluding Remarks and Future Research Directions 3 HEALTH CARE TEAM CAPACITY PLANNING 3.1 Introduction. 3.2 Literature Review. 3.3 Problem Formulation. 3.4 Queueing Analysis 3.4.1 Preemptive Case: Emergency Medicine Services 3.4.2 NonPreemptive Case: Outpatient Clinic Services 3.5 Computational Study. 3.5.1 Computational Performance of the DBA Method. 3.5.2 Computational Performance of the HCTSCP Model 3.6 Concluding Remarks and Future Research Directions 4 HOSPITAL BED ALLOCATION PROBLEM. 4.1 Introduction. 4.2 Literature Review. 4.3 Problem Formulation. 4.4 Solution Algorithms Ittering . rameters 4.4.1 Genetic Algorithm . ....... .... 75 4.4.2 Greedy Randomized Adaptive Search Procedure .. .. .. .. 80 4.4.3 Hybridization of GA & GRASP ..... .. . 81 4.5 Computational Study ........ . .. .. 83 4.5.1 Summary of Results Obtained by GA ... ... .. .. 85 4.5.2 Summary of Results Obtained by GRASP ... .. .. 88 4.5.3 Summary of Results Obtained by HA ... .. .. 88 4.6 Concluding Remarks and Future Research Directions .. .. .. 92 5 EMERGENCY ROOM SERVICES FACILITY LOCATION AND CAPACITY PLANNING ... ......... ........... 94 5.1 Introduction ......... . .. .. 94 5.2 Literature Review ......... .. .. 95 5.3 Problem Formulation ......... .. .. 97 5.4 Solution Approach ......... .. .. .. 100 5.4.1 Lagrangian Relaxation Approach .... ... .. 100 5.4.2 Lower Bound ......... .. .. 101 5.4.3 Upper Bound ......... .. .. 103 5.4.4 Lagfrangfian Multipliers . ...... .. 106 5.5 Computational Study ......... .. .. 107 5.5.1 Experimental Desigfn . .. .. 107 5.5.2 Experimental Results .. . .. 108 5.6 Concluding Remarks and Future Research Directions .. .. .. 111 6 CONCLUSIONS AND FUTURE RESEARCH DIRECTIONS .. .. .. .. 114 REFERENCES ......... . .. . 117 BIOGRAPHICAL SK(ETCH ....._._. .. .. 125 LIST OF TABLES page 27 27 :31 :32 Table 21 Parameter settings for the base scenario, S1 ......... 22 Scenario descriptions for experiment 1 ......... 23 Suninary statistics for the RBCPwS problent's solution time (in CPU seconds) as a function of initial effective capacity ......... 24 Suninary statistics for the RBCPwS problent's solution as a function of capacity levels and the length of the planning horizon .......... The possible transitions enter state (ni, n2,m) for the ED setting .... Computational requirement of the DBA, AGE, and GE methods ..... Relative and percentage error: preeniptive case ... Relative and percentage error: nonpreeniptive case .... Team configurations ...... Parameter settings ...... Impact of the fraction of class 1 patients on the similarity index ..... Impact of unit patient delay cost on the similarity index ..... . Impact of nmaxiniun allowable average time in system on the similarity in Nearoptinmal solutions obtained by using CPLEX ....... Solutions obtained by GA ...... Solutions obtained by GR ASP ....... Solutions obtained by HA ...... Experimental factor settings ....... Experiment 1: effects of nmaxiniun number of facilities opened ...... Experiment 2: effects of capacity setting ...... Experiment :3: effects of diversion probability ...... Experiment 4: effects of time value ...... Performance of the heuristic ... . 49 . 58 . 60 . 61 . 6:3 . 6:3 . 65 . 65 dex .. 65 . 86 .. 87 . 89 . 91 .. 107 .. 109 .... 110 .... 110 .. 111 S112 LIST OF FIGURES Figure 21 Network flow representation for RBCP with co=300, B=25, n=1, and T=4 page 21 22 23 24 25 31 32 41 42 43 44 45 46 47 48 49 410 411 412 413 51 52 53 54 55 Network flow representation for RBCPwS with co=275, B=25, Patient arrival rate for experiment 1 ..... Optimal capacity plans for experiment 1 ..... Number of nodes in the network as a function of initial effective An illustration of the network representation for HCTSCP .. Twodimensional CT3iC approximation ..... Pseudocode of the genetic algorithm ...... Pseudocode of the population generating procedure ...... Pseudocode of occupancydriven allocation ..... Pseudocode of randomrectified procedure ... Pseudocode of crossover procedure ... Example of crossover ...... Pseudocode of the mutation procedure ...... Pseudocode of GRASP ..... Pseudocode of greedy randomized construction procedure .. Pseudocode of local search procedure ... Pseudocode of HA ...... Pseudocode of elite set generation procedure ..... The updating functions of a ... Pseudocode of the Lagrangian relaxation ..... Pseudocode of the feasible solution generation ...... Pseudocode of covering all demand nodes ..... Pseudocode of closing facilities ..... Convergence speed of the modified LR ..... n=1, and T=4 . 28 . 28 e bed capacity 30 . . 47 52 . 76 .. 77 .. 77 . 78 . 79 . 79 80 80 . . 82 83 83 84 90 .. 101 .. 104 .. 105 .. 105 .. 109 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy OPTIMIZATION MODELS FOR CAPACITY PLANNING IN HEALTH CARE DELIVERY By ChinI Lin May 2008 C'I I!r: Elif Akgahl Major: Industrial and Systems Engineering Health care capacity planning is the art and science of predicting the quantity of resources required to deliver health care service at specified levels of cost and quality. Because of variability in the arrival of patients and in the delivery of health care services, successfully meeting the demand for health care services is a daunting task that requires an understanding of the inherent tradeoff between its cost and quality of service. In our work, we model the general health care systems as queueing stations and incorporate queueing theory into an optimization framework. The queueing modeling approach captures the stochastic nature of arrivals and service times that is typical in health care systems. The optimization framework determines the minimum cost capacity required to achieve a target level of customer service. The inclusions of queueing equations and discrete capacity options result the capacity planning models in nonlinear integer programming formulations. We develop effective solution algorithms to obtain high quality solutions particularly for realisticsized problems. For the analysis of underlying queuing systems, we either use available results from the literature or develop approximations. For the solution of optimization models, we employ network optimization, metaheuristic, and Lagrangian relaxation approaches to develop effective solution algorithms. We present results from extensive computational experiments to demonstrate the computational efficiency and effectiveness of the proposed solution approaches. CHAPTER 1 INTRODUCTION Capacity planning decisions are important to any industry, especially to health care industry because not only it relates to the management of highly specialized and costly resources (i.e., nurses, doctors, and advanced medical equipment), but also it makes a difference between life and death in critical conditions. Health care capacity planning involves predicting the quantity and particular attributes of resources required to deliver health care service at specified levels of cost and quality. According to the resource availabilities, capacity planning can be classified into three levels including strategic, tactical and operational levels. In the strategic level, decision makers focus on the long term capacity decisions such as locations of medical facilities or sizes of medical facilities and workforce. The tactical level capacity plan concerns policies which could improve the service performance, on the perspective of either health care providers or receivers, by re I11.~ n .111. expanding or downsizing the current resources. In contrast, the capacity plan on the operational level concentrates on how to meet demand by using the existing resources through appropriate methods such as scheduling or overrun. In our study, we focus on longterm level health care capacity planning decisions, where the knowledge of the performance of the system at steady state is sufficient. In addition, we model the health care service facilities as queueing systems to take the stochastic patient arrivals and length of stay into account and measure the system performance. The purpose of our study is to incorporate queueing models into optimization framework to determine the optimal capacity level that minimizes cost while maintaining a desired level of performance on patients' service quality and financial and/or operational restrictions. In our work, we study four different, practically relevant, longterm capacity planning problems. C'!. Ilter 2 introduces the aggregate hospital bed ** i''r ;1 / rher':..t. (AHBCP) problem, in which we model the hospital as a G/G/c queueing system with a single bed type and a single patient class. Using the approximation from Bitran and Tirupati [16, 17] to estimate the patients' expected waiting time, we determine the optimal bed capacity plan over a finite planning horizon utilizing a network flow approach. C'!s Ilter 3 presents the health care tersam r'/' I 1'1.r t. e.. t. (HC TCP) problem, which finds applications in hospital emergency room and outpatient clinical settings. The underlying queueing system is more complex than the one used for AHBCP. In particular, we consider a system with two classes of patients with different priorities and two types of health care teams with patient classdependent service rates. Moreover, there is o t;,l,,l,, I,.n, ;, ? 1.1 1.76.7.1 / between the teams, i.e., while one team can provide service to both classes of patient, whereas the other team can serve only one class of patient. For this queueing system, we develop an approximation approach to compute the average time that a patient spends in the system for each patient class. Then, we integrate the results from approximation method into an optimization model to make longterm health care team capacity decisions. C'!s Ilter 4 states the hospital bed allocation (HBA) problem, which is an extension the AHBCP problem. After the .I negate bed capacity is specified, the next step involved is concerned with the allocation of .l__a egate bed capacity among different medical care units (1!CI~s). We model each MCIT in a hospital as a Af/Af/c/c queueing system to estimate the probability of rejection when there are c beds in the unit. We develop an optimization model to allocate the .I negate bed capacity across different MCI~s. O nI Ilpter 5 details the emer i. ,:..;; room services f r. .:1.:1 location and r'/' I 1'7' : ': ': (ERSFLCP) problem, in which each facility is modeled as a Af/Af/c/c queueing system. We construct a facility location model, which simultaneously determines the number of facilities opened and their respective locations as well as the capacity levels of the facilities so that the probability that all servers in a facility are busy does not exceed a predetermined level. Last, C'!s Ilter 6 provides a summary of the four problems we have investigated and 0 _ r future research directions. CHAPTER 2 AGGREGATE HOSPITAL BED CAPACITY PLANNING 2.1 Introduction Capacity planning is central to the pursuit of balancing the quality of health care delivered with the cost of providing that care. Such planning involves predicting the quantity and particular attributes of resources required to deliver health care service at specified levels of cost and quality. In general, successful health care capacity planning must address a variety of issues, including the duration of the planning horizon (i.e., operational, tactical, and strategic), the level of care provided (i.e., primary, secondary, and tertiary), the type of care (i.e., inpatient and/or outpatient), the amount, capability, cost, and types of available or desired resources (i.e., doctors, nurses, technicians, medical and clinical support staff, facilities including buildings, rooms, beds, parking spaces, medical diagnostic and monitoring equipment, or any other element that constitutes an "input" to the delivery of health care) as well as the customer service metrics or performance measures expected for the facility (e.g., patient length of stay, likelihood of full capacity where all inpatient beds or examining rooms are occupied, utilization of providers and facilities, and financial performance such as having expenses within or below budget). While capacity planning has challenged health care decision makers and researchers for decades [90, 91, 101], there is a renewed sense of urgency to address this problem. In addition to the perennial struggle between the continually increasing costs of highly specialized and scarce inputs (i.e., skilled and flexible staff, advanced clinical and medical technology and equipment, physical space and supplies) and declining government and private reimbursements [89, 96], the demand for inpatient care has been growing substantially. According to the American Hospital Association (AHA), while average length of stay (ALOS) remained unchanged at 5.7 doi~ all community hospital volume statistics increased from 2002 to 2003: inpatient admissions by 0.C' to 34.8 million, total hospitalbased outpatient visits by 1."' to 563.2 million, emergency department visits by 1.0' to 111.1 million, adjusted average daily census (i.e., average number of inpatients and outpatients receiving care per d .;) by 0.C' to 894,000, and average inpatient occupancy rate increased by 1.C,' to 66.>' [7]. However, the number of hospitals of all types decreased by 30 to 5,764, there were 32 fewer community hospitals, and 8,000 fewer community hospital beds in 2003 [7]. In this paper, we focus on .l__ regate hospital bed capacity planning decisions. We develop a model to simultaneously determine the timing and magnitude of changes in bed capacity that minimizes capacity cost (including the cost of changing capacity as well as the cost of operating capacity) while maintaining a desired level of facility performance (e.g., limiting a patient's expected delay before being admitted to a bed and keeping expenses within budget) over a finite planning horizon. We divide the planning horizon into discrete time periods of equal length, and assume that the system achieves steady state in each of these intervals. This allows us to use queuing methodology to analyze system performance, but this typically leads to nonlinear equations in our formulation. As hospital bed capacity must be integer valued, our planning model is a largescale nonlinear integer optimization model that minimizes total cost while achieving a targeted level of system performance. We show that some practical considerations lead to simplifications in the model, which leads to a network flow formulation for the problem that can be solved in polynomial time. A variety of problems that arise in the context of transportation, finance, ]rn Ilrirll. (ilirin and service systems can be modeled as network flow models [2]. A network is a collection of capacitatedd or uncapacitated) nodes and (directed/undirected and capacitated/uncapacitated) arcs, where the arcs link one node to another and carry flow from one node to another. Wellknown network flow models are the shortest path, maximum flow, and minimum total cost flow formulations, for which efficient solution algorithms exist [2]. In our work, we show that the capacity planning model we consider can be transformed into a shortest path model, where the objective is to find the path from the source node to the sink node with the shortest length (i.e., the minimum cost bed capacity plan from the current period to the final period of a given planning horizon). The remainder of this paper is organized as follows. Section 2.2 provides a brief overview of the history and current research in hospital bed planning. In Section 2.3, we describe the system and give three models for planning hospital bed capacity. In Section 2.4, using data from a mediumsized medical center, we provide a computational study to illustrate how the model formulations can he used and how changes in problem parameters can affect our ability to obtain an optimal solution. Section 2.5, offers several practical extensions of our model. Last, we give concluding remarks and discuss future research directions in Section 2.6. 2.2 Literature Review During the 1990s, many hospitals in the United States reported having too many beds and were exploring strategies to reduce space [10, 15, 29, 45, 46, 50]. Less than a decade later, in part due to renewed growth in demand for inpatient services [7, :31], most hospitals are currently facing considerable space and resource restrictions forcing them to contemplate expensive renovations and/or new construction projects to increase bed capacity [11, :31, 57]. However, whether hospitals, in fact, need the additional capacity appears to be unresolved [10, 45]. On one hand, increased inpatient admissions coupled with fewer hospitals and fewer hospital beds would support the argument in favor of capacity increases [7, :31]. Conversely, level or decreasing average length of stay and a corresponding decrease in the average inpatient occupancy rate may imply that existing capacity is sufficient [10]. Regardless, determining the optimal number and organization of hospital beds continues to be a challenge. The ability to anticipate bed demand and match it with the appropriate bed supply is critical to effective bed planning. Health care decision makers know that both will be influenced by a number of factors. Factors internal to the decision makers include containing the costs associated with operating, contracting, and expanding current bed capacity, reducing bed assignment waiting, maintaining quality of care when patients are placed in inappropriate units (e.g., an intensive care patient may have to be placed in a cardiac unit), eliminating emergency department bottlenecks (i.e., keeping patients in the emergency department after initial treatment due to unavailability of beds in the appropriate care unit), and reducing the probability of diverting patients to other hospitals due to lack of bed capacity [45, 46]. Externally, factors facing decisionmakers include atypical changes in community health (e.g., severe flu strains), annual Is id .~l17< (e.g., Thanksgiving), and the availability, size, and composition of appropriate medical personnel. Historically, starting with the HillBurton Act of 1946, bed capacity planning has tended to be based on target occupancy levels (TOLs) that are assumed to reflect capacity levels that achieve an appropriate balance of cost, patient d. 1 .1 and resource utilization. TOLs are derived using analytic models of typical hospitals in different categories and are based on acceptable patient d. 1 .1< for different services. However, Green and Nguyen [46] use queuing models to investigate the relationship between occupancy levels and delay, and concluded that using TOLs as the primary determinant of bed capacity is inadequate and may lead to excessive d. 1 .1< for beds. In particular, a TOL does not necessarily correspond to a desired service level, and there is a need to quantify the desired service level and measure its cost implications accurately. Ryan [95] provides a capacity expansion model with exponential demand and continuous time intervals and continuous facility sizes. In the context of health care pl1 ....lr however, it is more realistic to model capacity expansion as the product of limited, discrete choices as routine planning sessions (e.g., bimonthly or quarterly) where capacity increases or decreases occur in some fixed bed amount such as a 20bed unit. Bretthauer and Ci~ti [26] model a general health care delivery system as a network of queuing stations and incorporate the queuing network into an optimization framework to determine the optimal capacity levels subject to a specified level of system performance (e.g., average total time spent at the facility). They use an algorithm combining branchandhound with outer approximation cutting plane method to solve the nonlinear optimization problem with discrete variables, but a disadvantage of this algorithm is that in the worst case the algorithm could require complete enumeration of all integer solutions, leading to very large solution times. 2.3 Problem Formulation In the bed capacity planning problem, we start with a planning horizon of length T indexed by t=1, 2, ..., T. Let A, denote the .I__regate patient arrival rate in period t, 1/p he the ALOS per patient, and the service rate per bed per dwi is given by ft or 1/ALOS and the service rate per bed over period t is ys,. In practice, there are alternative patient streams (including admissions front the emergency department, admissions front referrals, and elective admissions) for each of which the typical length of stay may be different. As the objective of our work is to provide an .I__aegate planning tool for bed capacity nianagenient, we assume that the arrival rates for different patient streams can he combined and a representative value for the average length of stay per patient (regardless type of services required by the patient) can he determined. Note that while ALOS has been relatively stable over time [7], the actual A, for a given facility will not he known until the demand presents itself. Therefore, for the purposes of capacity planning, A, can he forecasted by a seasonally adjusted trendline, for example [32]. Let nt denote the nmaxiniun allowable expected delay for a patient before the patient is admitted to a bed in period t. We note that the number of beds in the system in a given period can he limited due to other resource limitations including as the physical size of the facility and/or the amount and type of personnel available. Let co be the initial bed capacity in the hospital. Last, there is a budget limit on the amount of monetary resources that can he allocated to purchasing additional bed capacity denoted by yt. We have three types of decision variables. Let .r, he number of beds in period t. Let .r,+ he the amount of increase in bed capacity at the beginning of period t, and r, the amount of decrease in bed capacity at the beginning of period t. Let f (xt, Xt, It) denote the expected patient waiting cost as a function of number of beds xt, patient arrival rate Xt, and average service rate Pt in period t. Similarly, let g(xt_l, xt) denote the cost of changing bed capacity from xt1 to xt (i.e., the cost of increasing or decreasing the existing bed capacity) in period t. Let h(xt) denote the cost of operating xt beds in period t. Finally, the expected d+. i for a patient in period t is a function of number of beds xt, patient arrival rate Xt, and service rate per bed pt, denoted by w(xt, Xt, It). We can formulate the ..:o negate hospital bed capacity planning (AHBCP) problem as a nonlinear integer programming formulation as follows: I~i ? )fZ!LT T T) t= 1 t= 1 t= 1 subject to w(xt, Xt, It)l < t Vt (22) Xo = co (23) xt1 + x, xt = xt Vt (24) g(xt1, xt)l The objective function (21) minimizes the total cost of patient waiting, changing the bed capacity, and operating the existing bed capacity. Constraint (22) imposes a maximum allowable limit on the expected patient waiting. For example, in order to quantify the expected delay for a patient to be admitted to a bed, we assume that the hospital can be represented as a GI/G/s queueing system and use the expected waiting time approximation provided by Bitran and Tirupati [16, 17] to calculate a patient's expected wait for a hospital bed. Constraint (23) sets the initial bed capacity while constraint (24) is a flow balance equation stating that the number of beds available in a period is equal to the number of beds available in the previous period plus the increase in bed capacity minus the decrease in bed capacity. Constraint (25) is the budget constraint that limits the amount of the funds allocated to changing capacity. Last, constraint (26) ensures that the number of beds available and changes in bed capacity are integer valued. 2.3.1 Restricted Bed Capacity Planning Problem It should be readily apparent that the number of integer variables associated with the AHBCP problem could be quite large as there is no restriction on how many beds can be added or removed from service. For example, community hospitals may have 500 or more beds [7]. In practice, bed capacity is increased or decreased in batches, and is typically changed in integer multiples of a base value, or, in multiples of 10 or 25 corresponding to the size of a unit. As a result, there are only a limited number of choices for changing capacity in each period. Therefore, constraints that capture the change in capacity can be replaced by a set of discrete alternative constraints, requiring that only one alternative is chosen in the solution for each period. Then, the original nonlinear integer programming problem becomes a nonlinear binary (i.e., zeroone) integer programming problem, which we refer to as the restricted bed capacity planning (RBCP) problem. In the RBCP problem, we are given a base value of B in multiples of which the bed capacity can be increased or decreased and we let a be the number of possible distinct levels of capacity increase or decrease. That is, given bed capacity c in period t, the bed capacity in period t + 1 can be one of (c nB)+, (c (n 1)B)+, ..., (c B)+, c, c + B, ... c +(n 1)B, c +nB, where (x)+ = max{0, x}. We assume that all acquired new additional capacity is available and becomes effective capacity in the same period. Let zgt = 1 if the available bed capacity is increased by iB at the beginning of period t for i=1, 2, ..., n; and 0 otherwise. Similarly, let zy = 1 if the bed capacity is decreased by iB at the beginning of period t for i=1, 2, ..., n; and 0 otherwise. We can now formulate the RBCP problem as a nonlinear zeroone integer programming problem as follows: T T T minl Cf (xt, At, Pt) + g(xt1, ) + h(xt) (2 7) t= 1 t= 1 t= 1 subject to w(xt, Xt, It)l < t Vt (28) Xo = co (29) xt1 + i~zg i :r t V~t (210) i= 1 i= 1 zg a+ zy < 1 Vt (211) i= 1 i= 1 g (xt1, xt)l zg, g e 0, } V t(214) As in the AHBCP problem, objective function (27) minimizes the total cost of patient delay, changing the bed capacity, and operating the existing bed capacity, constraint (28) imposes a maximum allowable limit on the expected patient delay, constraint (29) sets the initial bed capacity, and constraint (210) is a flow balance equation. Constraint (211) ensures that only one choice for changing the capacity is allowed in each period. Constraint (212) imposes the budget constraint on the amount of money allocated to changing bed capacity. Constraints (213) and (214) ensure the nonnegativity of the bed capacity level and capacity level selection decision variables, respectively. An attractive feature of the RBCP problem is that a network representation can be developed. Consider a Tpartite graph with T lI;rs each representing a time period t=1, 2, ..., T in the planning horizon. Let (t, c) denote the system when there are c beds in period t. Let C(c) be the set of reachable capacity levels in the next period if the capacity in the current period is c, and we have C(c) = {(c nB)+, (c (n 1)B)+,..., (c  B)+, c, c + B, ..., c + (n 1)B, c + nB}. Let St be the set of all capacity levels reachable in period t from all capacity levels in period t 1. Let d, be a superficial source node connected only to node (0, xo) with zero arc length. Node (0, xo) represents the beginning state where there are xo beds in the hospital at time t=0. Let (0, xo) be connected to all nodes (1,x') for x' E C(xo). If w(x', Az, pt) < atl (i.e., the patient waiting time constraint is not violated) and g(xo, x') < yl (i.e., the budget constraint is not violated), then the length of these arcs are given by f(x', Ai, pt) + g(xo, x') + h(x') (i.e., the expected patient waiting cost with x' beds in the system, total cost of changing the bed capacity from xo to x', and cost of operating x' beds). However, if either constraint is violated, then the length of the corresponding arc is set to M~, where M~ is a very large number. Similarly, let each node (t, x) for x E St and t=1, 2, ..., T1 be connected to (t + 1, x') for x' E C(x) with length f(x', Xt+l, Ii+i) + g(xt, x') + h(x') if w(x', Xt+l, I#<+l) < a~t+l and g(xt, x') < 7t+l, and M otherwise. Last, let each node (T, x) for x E ST be connected to a superficial sink node dt with an are of zero length. Figure 21 provides an example of the network representation for the RBCP problem where co=300, B=25, n=1, and T=4. In this figure, a path from the superficial source node to the superficial sink node represents a plan for the bed capacity over the planning horizon. The shortest path without containing any arc with cost M~ yields the capacity plan with total minimum cost that obeys the patient waiting time and budget constraints over the planning horizon. If no such path can be found (i.e., the shortest path contains at least one arc with cost M~), then the problem is infeasible and no capacity plan that obeys the waiting time and budget constraints over the planning horizon can be found. Figure 21. Network flow representation for RBCP with co=300, B=25, n=1, and T=4 Recalling that there are n distinct levels to increase or decrease capacity, the general network flow representation representation for the RBCP problem has 2nt + 1 nodes in 1.s. rt for t=1, 2, ..., T. Therefore, there are a total of nT(T + 1) + T + 2 nodes (including the superficial sink and source nodes) and the shortest path for the RBCP problem can be found in O(n2 4) time using Dijsktra's algorithm [2]. 2.3.2 Restricted Bed Capacity Planning Problem with Shuttering In the RBCP problem, we assume that the cost of increasing or decreasing bed capacity is uniform. In practice, however, decreasing bed capacity can be achieved by shuttering existing bed capacity. That is, a hospital unit is closed and the personnel may be reassigned to other units in the hospital or laid off, thereby, reducing the effective bed capacity. On the other hand, increases in bed capacity can be accomplished two v .1<. If the existing capacity is larger than the effective capacity implying that shuttered capacity is available, then restoring a shuttered unit into operation by reallocating personnel to this unit can increase bed capacity. However, if the existing capacity is equal to the effective capacity implying that no shuttered capacity is available, then bed capacity can only be increased through a capital investment to open a new unit and purchase new beds. We can incorporate this practical concern into our formulation easily by changing the definition of the objective function by keeping track of the effective and existing bed capacity in the hospital. We now distinguish between two types of capacity changes, where g(xolx', 2')> is the cost of changing effective capacity from xo to x' via shuttering and the existing bed capacity from x to x' via acquiring additional capacity where x' > max~x', x}. As before, we assume that all acquired new additional capacity becomes effective capacity in the same period. The formulation is still a nonlinear zeroone integer programming problem, which we refer to as the restricted bed capacity planning with shuttering (RBCPwS) problem. As with the RBCP problem, a network representation can be developed for the RBCPwS problem. Consider a Tpartite graph with T lIn;rs each representing a time period t=1, 2, ..., T in the planning horizon. Let (t, clc) denote an effective capacity of c and an existing capacity of c in time period t, and c < c. Let Cl(clc) denote the set of reachable capacity levels via shuttering only, C2(~clc) ~lthe set of rea~chalev capaicit~y levels by acquiring new additional capacity and C(clc) =Cl(clc) U C2(clc) lthe set~ of all reachable capacity levels in the next period if the effective and existing bed capacity in the current period are c and c, respectively. If we have c + nB < c, then we have C, (clc) = {(((c nB)+Ia), ..., ((c B)+Ia), (cl c), (c + BI c), ..., (c + nB I )} and C2 (C C> = (8. Also, if we have c < c < c+ nB, we have Cl(cl c) = {((c nB)*a)c,..., ((c B)*a)c, (clc), (c+ Blc),..., (clc)} and C2(clc) =( {(+ BI ),..., (c +nB I )}. Again, let d, be a superficial source node connected only to node (0, xo Ix) with zero arc length. Node (0, xo Ix) represents the beginning state where there are x beds in the system and xo in operating condition at t=0. Let (0, xo2) be connected to all nodes (1, x'2) in C(xolx). Provided both the patient waiting time constraint and the budget constraint are not violated, then the length of these arcs are given by f(x', Ai, pt) + g(xolx, x'2') + h(x') (i.e., the expected patient waiting cost with x' beds in the system, total cost of changing the effective bed capacity from xo to x' via shuttering and the existing bed capacity from x to x' via new bed acquisition, and cost of operating x' beds). If either of these constraints is violated, then the length of the corresponding arc is set to M~. Similarly, each node (t, x2) for (x2) E St and t=1, 2, ..., T1 be connected to all nodes (t +1, x'2') in C(x2') with length f(x', Xt+l, It+l)+g(x2, x'2')+h(x') if w(x', Xt+l, I#<+l) < a~t+l and g(xlx, x'2x') < 7t+l, and M~ otherwise. Finally, let each node (T, x2) for (x2) E St be connected to a superficial sink node dt with length zero. Figure 22 provides an example of the network representation for the RBCPwS problem where co=275, co=300, B=25, n=1, and T=4. For ease of exposition, the thin arcs represent opening, maintaining, or shuttering of existing capacity, whereas the thick arcs represent the acquisition of new capacity. In this network, a path from the superficial source node to the superficial sink node represents a plan for the bed capacity throughout the planning horizon. As before, the shortest path without containing any are with cost Af in the network yields the capacity plan with total nxininiun cost that obeys the patient waiting time and budget constraints throughout the planning horizon by allowing capacity changes via shuttering and/or acquiring additional capacity. If no such path can he found (i.e., the shortest path contains at least one are with cost Af), the problem is infeasible and no capacity plan that obeys the waiting time and budget constraints over the planning horizon can he found. Figure 22. Network flow representation for RBCPwS with co=275, B=25, n=1, and T=4 Recalling the RBCP problem, we have specified the number of arcs and nodes in the network to determine the time to obtain the optimal solution. However, for the RBCPwS problem, since existing capacity can he increased further through capital acquisition, the analysis becomes slightly more complicated and dependent on the initial state (i.e., the amount of effective and existing bed capacity). If no additional capacity has to be purchased throughout the planning horizon, then the RBCPwS and RBCP networks are identical and the size of the RBCP network is a lower bound on the size of the RBCPwS network. If additional capacity has to be purchased in a period, at most (t 1)n2 nodes can be added to the network in that period. Hence, if the initial effective capacity is equal to the existing capacity, then there can be at most a total of nT(T + 1) + T + 2 + T(T + 1)(T 1)n2/6 nodes (including the superficial sink and source nodes) in the network, and the shortest path can be found again in O(n4 6) time using Dijsktra's algorithm [2]. 2.4 Illustration of the Model In this section, we illustrate the practical applicability and computational behavior of our model through two experiments. In the first experiment, we illustrate how our model can be used to develop bed capacity plans. In the second experiment, we quantify the time (in CPU seconds) needed to obtain optimal solutions. In both experiments, we use the RBCPwS formulation and its associated network model. 2.4.1 A Representative DecisionMaking Scenario To set the stage for the computational experiments that follow, we present a representative decisionmaking scenario based upon a realworld application of our model to a mediumsized, nongovernment, notforprofit, general medical and surgical medical center. Administration at this facility provided us with information about their facility, capacity planning decisionmaking processes, and facili T specific data for bed size, bed operating cost, bed acquisition cost, and quarterly patient demand. However, note that at the request of the facility's administration, the data presented here have been modified to protect their identity, but are representative of similarsized facilities. This facility would like to determine an optimal bed capacity plan for the next eight quarters, corresponding to its operational, budgetary, and strategic planning periods. Because capacity planning may involve a substantial capital commitment, it is imperative that any capacity expansion plan be carefully developed and justified based upon the facility's current and expected demand. The facility's decision makers would like to minimize the total capacity cost associated with the cost of changing capacity as well as the cost of operating capacity while ensuring that the average time a patient should wait for a bed does not exceed one hour (an internal benchmark for bed assignment). At this facility, both existing and effective bed capacities are :350 heds, capacity change can occur in increments of 10 heds, and there are two levels of capacity increase (i.e., initially, bed capacity can range from :330 to :370 heds, in 10 hed increments). Based on information from the facility's administration, it costs $2,000/d is to operate an effective bed, $2,500/d is to either shutter an effective bed or reactivate a shuttered bed, and $200,000/hed to expand bed capacity through capital investment. Last, because of seasonal migration (or a! 0. hirds"), demand at the facility can be highly variable throughout the year, and we were provided with data and guidance on values related to patient arrival rates and service times. 2.4.2 Experiment 1 An Application of the Model The intent of this experiment is to illustrate how our network flow model can he used to make bed capacity decisions and generate a Tperiod capacity plan. Our base scenario was described in Section 2.4.1, and we refer to it as S1. Table 21 lists the relevant parameter settings for S1, and other experimental scenarios relative to this scenario are given in Table 22. At the outset, we provide an estimated range of demand for the facility over the planning horizon. Normally, a single seasonally adjusted trendline would be computed to forecast the patient arrival rate based on historic demand data. Instead, to illustrate the extent of variation in demand, Figure 23 di pF.T4~ a set of simulated patient arrival rates over the planning horizon based upon the scenarios given in Table 22. We note that some of the parameter changes directly impact the patient arrival rate, and different patient arrival rates are generated. In S1, S:3, S4, S5, and S6, the changed parameters do not impact the arrival rate function, so these scenarios have identical arrival rates. (Note Table 21. Parameter settings for the base scenario, S1 Parameter Value Length of the planning horizon T = 8 quarters, t= 1, 2, ..., 8 Forecasted demand per time period t Xt = Smod(t,4)U(a + bt) where as is a quarterly seasonal index (i.e., sl=0.8, s2=1.0, s3=1.2, and s4=1.0), a is a uniformly distributed random number (i.e., u~ U[0.8, 1.2]), a=6,400, and b=128 Initial existing bed capacity co = 350 Initial effective bed capacity c = 350 Number of levels of capacity increase or n = 2 decrease Incremental amount of capacity change Cost to operate an effective bed Cost to shutter an effective bed Cost to reactivate a shuttered bed Cost to acquire a new bed (i.e., expand capacity through capital investment ) Coefficient of variation for arrivals Coefficient of variation for service Maximum expected delay per patient Cost of waiting Service rate B =10 $2,000/bed $2,500/bed $2,500/bed $200,000/bed cat = 0.5 est = 0.5 at = 1 hour $300/hour It = 15.8 patients per bed Table 22. Scenario SO S1 S2 S3 S4 S5 S6 Scenario descriptions for experiment 1 Description Level demand Base scenario Increased rate of demand Higher demand variability Higher service variability Higher cost of waiting per patient Smaller maximum expected del li per patient Parameter change b = b = 256 cat = 2.0 est = 2.0 $1,200/hour a~t = 0.25 of an hour that with S3, higher variability in the arrival rate impacts the performance constraint for average waiting time, not the arrival rate function.) For SO and S2, the patient arrival rate function has no trend and a higher trend compared to S1, respectively. Hence, arrival rates generated for these scenarios are significantly different from each other and S1. We have implemented our network flow approach using the C++ programming language and solved for the scenarios using a personal computer with 3.0 GHz Pentium IV * SO 350 330 310 290 270 250 230 250 250 CS1 350 330 310 320 300 280 290 300 310 AS2 350 330 310 320 300 310 330 350 370 S3 350 330 310 320 300 280 300 310 320 KS4 350 330 310 320 300 280 300 300 320 + S5 350 330 310 330 310 290 300 310 320 S6 350 330 310 320 300 280 300 300 310 Time periods Figure 24. Optimal capacity plans for experiment 1 11000 j 9000 18 8000 ~i7000 .k~ 6000  S5000 $* SO 4000  I S1,S3,S4, S5, S6 A S2 3000I IIIIII 12 3 4 5 6 7 8 Time periods Figure 23. Patient arrival rate for experiment 1 processor and 512 MB R AM memory. We obtained the optimal solution for each scenario and the results are depicted in Figure 24, where each line represents the optimal capacity plan that corresponds to one of the seven scenarios. 375 35) S325 275 0 SS() S1 A S2  M 3 rc S4 + S5  S6 In considering Figure 24, we have the following observations. For S1, we first observe a general reduction in the bed capacity, then a gradual increase near the end of the planning horizon. The initial bed capacity seems to be higher than needed, and as a result, the bed capacity is reduced to reduce total costs over the planning horizon while maintaining the average waiting time constraint. Of course, when the demand increases due to the underlying trend, the bed capacity is increased. When demand is level as in SO, a lower envelope is formed relative to the base case (i.e., the bed capacity for SO is less than or equal to the base case). Similarly, with an increased rate of demand as in S2, an upper envelope is formed relative to the base case. With increased variation as in S3 and S4, the optimal capacity plans are similar to S1's capacity plan but tend to require higher capacity when the arrival rate is increasing. When the arrival rate increases in periods 6, 7, and 8, because the higher arrival variability and higher service variability affect the average waiting time constraint, more capacity is required to keep from violating this performance constraint. Likewise, with a higher cost of waiting per patient as in S5 or a tighter average waiting time performance constraint as in S6, the optimal capacity plans tend to require capacity slightly higher than the base case. Not surprisingly, the net result of this experiment indicates that optimal bed plans are driven substantially by changes in demand. While health care decision makers may not he able to affect overall demand for their services, if they can reduce variability in arrivals [78] or are willing to tolerate a less stringent performance constraint, less capacity will be required. 2.4.3 Experiment 2 Assessing the Impact of Problem Parameters As we have discussed earlier, an upper bound on the size of the network (i.e., number of nodes in the network) representing a problem instance of RBCPwS can he characterized in terms of the number of levels for capacity increase or decrease and the number of time periods in the planning horizon. The ratio of the effective bed capacity to existing bed capacity impacts the size of the network. The size of the network can also be used to quantify the computing time required to obtain the optimal solution. The time required to build the network and find the optimal solution may change as the number of levels increases, the planning horizon length increases, or the ratio of effective to existing bed capacity changes. In order to illustrate the change in computational time, this experiment has two parts: 1) the impact of effective to existing bed capacity and 2) the impact of changes to the number of levels of bed capacity and the length of the planning horizon. In the first part of this experiment, we fix the number of levels to vary bed capacity and the duration of the planning horizon in addition to some other problem parameters constant and examine the impact of different ratios of existing to effective bed capacity. Using the assumptions for the base case scenario, S1, from the previous experiment, we consider ten different levels of the effective bed capacity in the interval [260, 350]. We generated 30 random test instances for each of these levels and the summary results are provided in Figure 25 and Table 23. 600 500 8 400 S300 S200 S100 250 260 270 280 290 300 310 320 330 340 350 360 Initial number of effective beds Figure 25. Number of nodes in the network as a function of initial effective bed capacity In Figure 25 we depict the number of nodes in the network, and in Table 23 we report the time to build the network and time to obtain the solution for each level of the initial effective bed capacity. The number of nodes increases as the ratio of effective bed capacity to existing bed capacity approaches one, and this behavior is clearly depicted in Figure 25. However, in Table 23, we see that an increase in the size of the network increases the time to build the network only slightly, and its impact on the time to obtain the solution is almost negligible. Therefore, our solution method is robust to changes in the problem size that are induced by the initial effective bed capacity. Table 23. Summary statistics for the RBCPwS problem's solution time (in CPU seconds) as Initial level of effective bed capacity 260 270 280 290 300 310 320 330 340 350 a function of initial effective capacity Time (in CPU seconds) to Build the netowrk Obtain the solution Min. Avgf. la~x. Min. Avgf. la~x. 0.0 0.0 0.2 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.1 0.1 0.0 0.0 0.0 0.1 0.1 0.1 0.0 0.0 0.0 0.1 0.1 0.1 0.0 0.0 0.0 Total time (in CPU seconds) Min. Avgf. Alax. 0.0 0.0 0.2 0.0 0.0 0.1 0.0 0.0 0.1 0.0 0.0 0.1 0.0 0.0 0.1 0.0 0.0 0.1 0.0 0.0 0.1 0.0 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 For the second part of this experiment, we vary the number of levels to change bed capacity as well as the duration of the planning horizon. We consider four different levels to vary the bed capacity (i.e., n=2, 3, 4, 5) where capacity is increased in increments of B=10, and three different time horizons (i.e., T=8, 12, 16) that correspond to two, three, and fouryear planning horizons. Therefore, we have 12 settings in total. For each setting, we generated 30 random instances and for each of the instances, we also generated the effective bed capacity as a fraction of the existing bed capacity. The summary results for this experiment are provided in Table 24. From Table 24, as the number of levels of capacity change and the length of the planning horizon increases, the number of nodes in the network increases. The increase in the number of nodes impacts the total time required to obtain the optimal solution. However, a closer examination of the results reveals the increase in the number of nodes in the network has a direct impact on the time required to build the network, and has almost no impact on the time to obtain the solution. Only in the setting with the largest test instances (i.e., n=5 and T=16) do we observe an increase in the time to obtain the '00300C\C3,0000 00O~ iic3~00+ 0 0000+ 000~C~u5~ 0000000000C\C\C\C\C\ 0000000000 0000000000 0000000000 MC u OH \ CO30 cio o~ asluC~ C~3 Ln~OO ~C~u5 C~u5C~ 00 OOp 00 W~~ O~cu~i cu oo ooc~o cu cue, IICU~ u5 CHC\ oO cmei oO CUN C\IC\IC\IC\IC~)C~)C~)C~) 00000000 iiiiiiiiiiiiiiii C\1C~3~u5 C\1C~3~u5C\1C~3~u5 optimal solution. Even in that case, the nmaxiniun solution time is still less than a few seconds. Therefore, our solution method is robust to changes in the problem size induced by the number of levels of capacity change and the duration of the planning horizon. 2.5 Extensions In this section, we discuss several extensions to our model. These extensions may arise out of practical considerations associated with how our model addresses facility performance . In our model, we treat the performance constraints as a hard constraint. That is, if a particular capacity level violates the performance constraint, then a solution with that particular capacity level is not feasible, and is dropped front further consideration. However, the performance constraint can he modeled as a soft constraint where we can deliberately allow the violation of the performance constraint while incurring a penalty cost to be added to the objective function. We can justify this constraint by noting that lags typically exist between capacity levels so there might he periods of time where the facility is operating above its typical utilization and the capacity expansion cannot occur quickly enough to allow the organization to react to the change in demand. To illustrate how our model can he reformulated with the soft constraint, let i, b e the amount of the violation, s, he the amount of slack in the performance constraint, and xr(l ,) be thepenalty cost incurred for violating the performance constraint in period t. Then, considering the RBCP problem, objective function (27) would be replaced with: T T T T nxin f (.rXI,LI) A, p) g(. t1I .) + Ch(.r,) + x (r ) (215) t= 1 t= 1 t= 1 t= 1 Similarly, constraints (28) and (213) would be replaced with: w(.r,, A,, pt) ', + S, = nt Vt (216) ', = nmax {w(.rt, A,, p,) c0,, 0} V t (217) s, = nmax {0, w(.r,, A,, p,), 0} V t (218) .r,, '<,S 8 > 0 V t (219) It is easy to observe that this modified version of the RBCP problem can still be formulated and solved as a network flow problem. The variables i, and .st are calculated by constraints (217) and (218), respectively, once w(.r,, A,, pi) is known. The only modification of the network is to include the cost associated with violating the performance constraint. When evaluating a hospital, we recognize that the average waiting time to be assigned a bed or having expenses within budget are not the only metrics to assess facility performance. Indeed, it may be necessary to include measures for facility utilization, likelihood of patient diversion, and the like. Regardless, we note that more performance constraints can easily be added to the formulations for the BCP, RBCP, and RBCPwS problems. An increase in the number of performance constraints does not increase the time to obtain the solution significantly, as there is only a need to take these additional constraints into account in setting up the network and assigning a large are cost in case any of the constraints are violated. Therefore, our modeling approach is robust and additional constraints can he considered without increasing the complexity of the formulation significantly. 2.6 Concluding Remarks and Future Research Directions We have presented a network flow approach to optimize bed capacity planning decisions for hospitals. Our model incorporates the reasonable concerns associated with determining hospital bed size, such as a finite planning horizon, an upper bound on the average waiting time before a patient is admitted to a hospital bed, and a budget constraint that limits the amount of money that can he allocated to changing bed capacity. Further, our model accommodates capacity change through Ihot 1 fling as well as expansion of bed capacity through new capital investment. Our series of computational experiments illustrated both the ease of implementation of our model and the sensitivity of the computational time required to obtain the optimal solution to several problem parameters. We have also discussed extensions of our model in the form of soft performance constraints and multiple performance constraints. Our model is based on a generic view of a hospital where we have assumed that the demand (i.e., patient arrivals) and service (i.e., beds) components are homogeneous. From an .I__oegate planning perspective, such uniformity may be acceptable. However, in order to apply this research to operational decision support for health care delivery, there are additional avenues of research worth pursuing. First, if cost depends on all previous stages, for example, the cost of maintaining the beds depends not only on the number of beds but also the duration of the beds are in the system, then the number of vertices in the network will be exponential with respect to T and the optimal solution to the network will not be solved with a polynomial time algorithm. Consequently, alternative model formulations and solution techniques to determine the optimal bed plan would be necessary. Second, recognizing that hospital beds are not identical, facility capacity could be separated to distinguish the various specialties, with specialti specific demand rates, lengths of stay, and costs. In determining the average waiting time associated with being assigned a bed, we have used closedform approximations to calculate this statistic. Therefore, we are implicitly assuming that this general distribution accounts for different types of patients that require different types of hospitalbased health care. This may not necessarily be the case, and should be investigated further. Third, our work can be expanded to include multiple types of patients (e.g., electives, admissions coming through the emergency department, and referrals from physicians). Also, in estimating the cost of patient waiting, we assume that this cost is identical regardless of patient type. Clearly, for example, there should be different waiting costs associated placing an emergency department admission in an appropriate unit versus an inappropriate unit. As such, representations of patient waiting cost need to be developed in the presence of congested, heterogeneous resources. Fourth, the current form of our model does not account for the potential time delay that may exist between the decision to expand capacity and actually starting to use the new capacity. Our current model formulations would have to be amended to include the length of delay relative to a capacity expansion (e.g., if a capacity expansion requires k time periods and we need to use the capacity in period t, the decision to expand should occur on or before period t k) and reconciliation of multiple capacity expansions over the planning horizon (e.g., if a capacity expansion decision is made in period t, can the facility make another decision in subsequent periods until t + k when the earlier decision contes into effect). However, because these capacity expansion considerations would destroy the underlying polynonlially bounded network structure of the current model, other solution methodologies would have to be developed. Last, as evidenced by the current nurse shortage [3] and the ongoing debate regarding nursetopatient ratios [4], the ability to use physical capacity hinges upon the availability of suitable medical personnel. A natural extension of our model would be to incorporate workforce planning to simultaneously determine the quantity and composition of the health care resources to construct a comprehensive capacity plan. CHAPTER 3 HEALTH CARE TEAM CAPACITY PLANNING, 3.1 Introduction With a size of $1.9 trillion and growth rate of 7.9 percent [102], the health care industry accounts for the largest sector of the economy in the United States (US). Despite advances in medical technology and, thereby, the increasing use of medical diagnostic, monitoring, and treatment equipment, the health care industry is highly laborintensive. According to the US Department of Labor, the health care industry provided 13.5 million jobs in 2004, out of which 13.1 million jobs are for wage and salary workers and about 411,000 are for the selfemploiv 1 [107]. It follows that personnel wages and salaries account for the largest portion of the total expenditures for any health care facility. For instance, hospitals spend on average about 54 percent of all expenditures on wages and salaries [86]. Hence, health care personnel pI l ..... i.e., determining the appropriate mix of health care personnel, needed to provide safe, effective, timely, and costefficient service to patients [55], is an important problem. In practice, both in inpatient and outpatient facilities, a health care team, comprised of a group of health care personnel with different, and complementary, skill sets, provides health care services to individual patients [34]. The members of the team (e.g. physicians, physician's assistants, and registered nurses) are responsible to perform a set of tasks required for the diagnosis, monitoring, and treatment of the patients. Some additional tasks may have to be performed by other personnel (e.g., laboratory technicians, radiological technicians, and radiologists) who are not a part of the health care team but provide assistance for diagnosis and treatment. In the delivery of services by health care teams, the safety and effectiveness of the service is ensured by the appropriate selection of the service *'istl~.:.71;i of the team, whereas the timeliness and costeffectiveness of the service is ensured by the appropriate selection of the service *r'ist..Uti of the team. The service capability of a team is characterized by the collection of the skills possessed by each of the individual members of the team, whereas the service capacity of the team is given by the total number of members included in the team. In a health care facility, there are typically multiple types of health care teams each with different capabilities. The patients that arrive to the facility are classified according to their conditions (i.e., acuity levels) or medical requests. Based on this classification, e ach p at ient i s as si gne d t o a health h care t eam and admit t ed t o an examinat ion /tre atment (E/T) room that has the equipment necessary to provide the service needed by the patient. As the set of tasks are shared among the members of the team and a typical facility has multiple E/T rooms, a health care team usually serves multiple patients simultaneously. For instance, while the team waits for the test results from the lab for a patient, a registered nurse may be collecting specimens from another, a physician's assistant may be suturing a wound of another, and a physician accompanied by a registered nurse may be discussing a treatment plan with another. In our work, we consider two particular settings where health care services are provided by teams. Shands at Alachua General Hospital in Gainesville, Florida is a community hospital that provides emergency medicine services. triage nurse, who determines the acuity level of the patient and identifies whether the patient requires immediate (i.e., emergency) or d.l I we 4 (i.e., urgent) care. The emergency care services are delivered by an emergency care (EC) team, which is composed of physicians and registered nurses, and the urgent care services are delivered by an urgent care (UC) team, which is also composed of physicians and registered nurses. We note that the triage nurse does not belong to either of these teams, but acts as a gatekeeper to route an arriving patient to either of the teams. There are eight and two E/T rooms dedicated to the EC and UC teams, respectively. Moreover, the EC team has the capability to attend to urgent care patients, but the UJC team does not have the capability to attend to emergency care patients. Finally, emergency care patients have preemptive Jp..r,:70l over urgent care patients. That is, if an emergency care patient arrives to the ED, while all E/T rooms dedicated to the EC team are occupied by other patients and one of them is an urgent care patient, then the emergency care patient preempts the urgent care patient out of the room and the emergency care patient is immediately admitted to the room. The Women's Clinic at the Student Health Care Center at the University of Florida in Gainesville, Florida provides women's health care services. The outpatient clinic (OC) serves not only nonacute patients, i.e., those who need routine services, but treats acute patients also. The services for the diagnosis and treatment of acute illnesses and abnormalities are provided by a physician (P) team, which is composed of a physician and a physician's assistant. The routine clinical services are delivered by a nurse practitioner (NP) team, which is composed of a nurse practitioner and a registered nurse. There are four E/T rooms dedicated to the NP team and two rooms to the P team. Moreover, the P team has the capability to attend to nonacute patients but the NP team does not have the capability to attend to acute patients. Finally, acute patients have nonpreemp~tive Jp.. r,:70 over the nonacute patients. That is, if an acute patient arrives to the OC, while all E/T rooms dedicated to the P team are occupied by other patients and one of them is a nonacute patient, then the P team does not interrupt service to the nonacute patient and the acute patient has to wait until an E/T room dedicated to the P team becomes available. In the settings we discussed above, the service capabilities of the health care teams are fixed as dictated by the service needs of the patients, and personnel planning is mainly concerned with determining the service capacity of the teams. In determining the service capacity, however, administrators must take several additional facility capacity, budgetary, and legislative constraints into account that limit the minimum and maximum total number of each personnel type emploi. I For instance, budget constraints may limit the total number of physicians emploi II whereas legislated nursetopatient staffing ratios may prescrobe a lower limit on the total number of registered nurses. Therefore, given the lower and upper bounds on the total number of personnel with different skills that can be emploi II there is a finite number of team configurations that can he utilized by a health care facility. For example, consider a health care team for which physicians and registered nurses are required. Suppose that due to a budget constraint, the health care facility can employ at most two physicians and four registered nurses at a time. Also, suppose that due to a legislative constraint, the facility should employ at least one physician and two registered nurses. Then, for this team, there are at most six feasible configurations, i.e., { (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4) }, that administrators can choose from. Therefore, in such settings, personnel planning is concerned with choosing an appropriate configuration for each type of health care team. In this paper, motivated by the practical settings we discussed above, we address the longterm health care team service capacity planning (HCTSCP) problem in the context of health care facilities where there are two patient classes and two types of teams. We assume that the capability of each type of team is known, and the number of E/T rooms allocated to each type of team is fixed over the planning horizon. Therefore, the service capacity of a team can only be changed by modifying the configuration of the team. 1\oreover, the service capacity of a team can he quantified by the service rate of the E/T rooms dedicated to the team, which is the reciprocal of the average time that the patients spend in the E/T rooms prior to discharge or transfer to another department. We formulate a nonlinear binary integer programming model to determine the service capacity plan for the health care teams such that health care services are delivered in a timely (i.e., the average time a patient from a particular class spends in the system does not exceed a prespecified threshold) and costeffective (i.e., the total costs associated with changing service capacity by hiring additional personnel, reassigning existing personnel or laying off existing personnel and operating the service capacity are minimized) manner over a given planning horizon considering some additional constraints. To estimate the average time a patient from a particular class spends in the system, we develop queuing models and decomposition based approximation (DBA) methods. We note that in the practical settings we consider the steady state results obtained from queuing analysis can be used in capacity pI ll onir_ as both systems are highly utilized, and, hence, the systems reach to steady state quickly in both of the settings. To model these settings, we consider queueing systems where there are two classes of patients and two types of teams. Each team has a set of of E/T rooms dedicated to it, and there is .Iiinin.! it ic substitutability between the teams and their dedicated E/T rooms. That is, class 1 patients (i.e., emergency care patients in the ED and acute patients in the OC) can only be served in the E/T rooms dedicated to type 1 team (i.e., EC team in the ED and P team in the OC), class 2 patients (i.e., urgent care patients in the ED and nonacute patients in the OC) can be served in the E/T rooms dedicated to either type 1 or type 2 team. An arriving class 2 patient is admitted to a vacant type 1 E/T room only when no class 1 patient waits in the system and no type 2 E/T room is available. In addition, service rates are patientclass and teamtype dependent, i.e., the time required to serve a patient depends on both the class of the patient being served and the type of the team that delivers the service. We consider both the preemptive and nonpreemptive cases. Our computational results illustrate that the DBA method and the capacity planning model can effectively be used to make longterm personnel planning decisions. The remainder of this paper is organized as follows. In Section 3.2, we review the related literature. Section 3.3 presents our capacity planning model for HCTSCP and show how it can be interpreted as a network flow model. We develop queueing models for the ED and OC settings and present DBA methods to analyze these models in Section 3.4. In Section 3.5, we present results from our computational study that evaluates the accuracy of the approximations from Section 3.4 and investigate the computational performance of the formulation presented in Section 3.3. Section 3.6 includes a discussion of the results and so__~ is future research directions. 3.2 Literature Review Health care personnel capacity planning has been been studied for decades to address longternt budgeting, niediumternt r 11lo and shortternt scheduling decisions. A critical examination of the analytical literature to date illustrates that a considerable amount of effort has been allocated to the shortternt personnel planning decisions. In particular, the nurse rostering (also known as the nurse scheduling or the nurse staffing) problem has been studied extensively, as nurse staffing costs account for a significant portion of personnel costs in health care system [61]. Burke et al. [28] provide an excellent review of the existing work in this area. Analytical work that focuses on niedium to longternt personnel planning is relatively limited in scope. Schneider and K~ilpatrick [97] develop optimization models for personnel planning in health care facilities. K~ao and Tung [62] present a linear progranining model for the .I negate (nursing) workforce planning problem, which is later extended by Brusco and Showalter [27] to account for the exogenous impact of nursing shortage. K~ropp and Carlson [72] propose the integrated use of optimization and simulation modeling. Existing work that uses simulation modeling is reviewed in Jun et al. [58]. Although the earlier work in this area primarily focuses on the perspective of the health care provider by placing an emphasis on nxinintizing the cost of personnel resources or nmaxintizing the utilization of personnel resources, there is a need to take the patient's perspective into account by considering the timeliness (e.g., nxinintizing of the waiting time and/or time in system) or the availability (e.g., nxinintizing the probability of finding all servers busy and being diverted to another service provider). To this end, we focus on nxinintizing the sunt of capacity costs and cost of not providing service in a timely manner, while ensuring that the average time in system for a patient does not exceed a prespecified threshold. To this end, we divide the planning horizon into discrete time periods of equal length and assume that the system achieves steady state in each of these intervals. This allows us to use queueing analysis to capture the stochastic behavior of the system and compute the average time in system for the patients. Using the results of this analysis, we formulate the HCTSCP problem as a mathematical programming model as in [5]. In the literature, there is a number of studies that investigate queueing systems that are closely related to the setting we consider. Stanford and Grassmann [105] derive the expected waiting time in a call center with unilingfual and bilingual servers serving ill ll .s Ry and nrlrairs..vl I1exuageuse customers. A nrrairs..vi( i.1 .vs uageuse customer can only be served by a bilingual server, and the type of a customer is not be known prior to the first service. The service rates for all server types are the same, i.e., independent of the customer type. Green [44] and Shumsky [99] also investigate expected waiting time of a system with two types of customers and limited and generaluse servers. Both of them consider the case where the service rates depend on the type of the server. Hence, the distinguishing characteristic of the system we investigate is the dependence of the service time on customer class for one of the server types. This seemingly a simple attribute of the system leads to significant modeling challenges. In queueing theory, to obtain the stationary probabilities of a system, the 1 its lin approaches used include the matrix analytic methods [44, 60, 74, 75, 87, 105], powerseries algorithm (PSA) approach [19, 20, 54, 71], and DBA method [99]. The matrix analytic methods formulate the system as a Markov chain to which the stationary probability xr has the matrixgeometric form ar = xroR", where rate matrix R can be obtained through an iterative algorithm [87]. The PSA approach first represents the stationary probabilities of a system as powerseries expansions of the traffic intensity of the system, and then recursively solves for the coefficients of the powerseries expansions by using the set of stationary equations. K~ao and Wilson [64] compare the performance of the PSA and matrix analytic methods with three iterative algorithms proposed in [60, 74, 75], and conclude that the PSA performs extremely well in terms of computational speed, though it may encounter difficulties in parameter settings which may lead to losses in accuracy. Shumsky [99] proposes the DBA method. The DBA method first divides the state space of the system into several regions, and then estimates the stationary probabilities in each region by using simple approximate queueing models. Shumsky [99] illustrates that this method generates performance measures rapidly with sufficient accuracy that can be used in call center capacity decisions.In our queueing analysis, we also use this DBA method. 3.3 Problem Formulation In the HCTSCP problem, we are given a planning horizon of length T, indexed by t = 1, .. ., T. In each planning period, there is a limit on the funds that can be allocated to changing service capacity of the teams in the facility, denoted by 7t for t = 1,..., T. There are two patient classes, indexed by i = 1, 2. For each patient class there is an upper limit on the number of patients in the facility, denoted by bi, and let b be the twodimensional vector representing these limits. In addition, there is an upper bound on the average amount of time that a class i patient spends in the facility, denoted by asi, for i = 1, 2. The forecasted arrival rate of class i patients in period t is denoted by Xie for i = 1, 2 and t = 1,..., T, and let Xt be a twodimensional vector associated with the arrival rates of patients in both classes in period t. There are two types of health care teams, indexed by j = 1, 2, and two types of E/T rooms. Team type 1 can provide service to both patient classes with different service rates in its dedicated E/T rooms, type 1, while team type 2 can only provide service to patient class 2 in its dedicated E/T rooms, type 2. Let rj denote the number of E/T rooms allocated to team type j for j = 1, 2, and r be the associated twodimensional vector representing the room allocation. As we discussed earlier, the service capacity of each type of team can only be changed by modifying the configuration of the team and can be quantified by the service rates of the associated E/T rooms. Given the lower and upper bounds on the number of personnel with different skills that can be emploix & in the facility and the types of personnel that must be included in a particular type of team, let Ky denote the total number of possible configurations for team type j, indexed by k = 1,...,Ky. Suppose that it is possible for the administrators to estimate the service rate 8ijk, of type j E/T rooms when the associated team, i.e., team type j, has configuration k, and the patient in the room belongs to class i for i = 1, 2, j = 1, 2 and k = 1, .. ., Ky. We note that 012k = 0 for k = 1, K2 Since the class 1 patients cannot be treated by the type 2 team or in the type 2 E/T rooms. We have two sets of decision variables. Let xjkt take the value of one if for team type j configuration k is selected in period t for j = 1,2, k = 1,...,Ky and t = 1,...,T, and zero otherwise. Also, let xjt be a Kydimensional vector associated with the team configuration decision for team type j in period t. Finally, let '., denote the service rate for class i patients who are treated in type j E/T rooms in period t for i = 1, 2, j = 1, 2 and t = 1,..., T, and 4t be a 2 x 2 matrix associated with the service rates of the E/T rooms in period t. We let I, (#t, Xt, b, r) and as(#t, Xt, b, r) represent the waiting cost of class i patients and the average time spent in the system by class i patients, respectively, as a function of the service rate of the E/T rooms ~t in period t, patient arrival rates At in period t, maximum number of patients allowed in the facility b and E/T room allocation r. Also, we let cj (xjt1, xjt) denote the cost of modifying the configuration of type j team from period t 1 to t and oy (xjt) denote the cost of employing the personnel necessary for the chosen configuration for team type j in period t. Assuming that all acquired additional personnel capacity is available and becomes effective in the same period, the HCTSCP problem can be formulated as a nonlinear binary integer programming problem as follows: i= 1 t= 1 j= 1 t= 1 j= 1 t= 1 subject to xjkt= 1 j, t(32) ~ Oij 3k kt = 0 i j (33) si(#t, Xt, b, r)l < a V, (34) cy (xt1,xat)< TeVt (35) j= 1 Zjkt E {0, 1} j k (36) The objective function (31) minimizes the sum of the cost associated with the time that the patients spend in the system, the cost of modifying team configurations to change the service capacity of the teams, i.e., service rates of the associated E/T rooms, and the cost of employing the personnel necessary for the selected team configuration. Constraints (32) stipulate that one configuration must be selected for each team type in each planning period. Constraints (33) assign the service rates for E/T rooms according to the selected configurations for each team type with respect to different patient classes. Note that (12t = 0 for t = 1,..., T due to 012k = 0 for k = 1,... ,K2. COllStraintS (34) impose upper limits on the average time that patients spend in the system for each patient class. Constraints (35) limits the amount of funds that can be allocated to changing team configuration in each planning period. Finally, constraints (36) and (37) ensure the integrality of team configuration and nonnegativity of service rate decision variables, respectively. HCTSCP is a difficult nonlinear binary integer programming problem with nonlinear constraints. Note that, however, HCTSCP can be represented by a T + 1partite network, where each l~i;r in the network represents a time period t = 0,..., T in the planning horizon. Let (kl, k2, t) denote the facility when configuration kgi for type j team is used in period t. L ... rrt = 0 include a single node (kl, k2, 0), which denotes the initial configurations for type 1 and type 2 teams. A superficial source node S is connected to node (kl, k2,0O) only with zero arc cost. Each 1... rt = 1,..., T contains K1 x K2 nodes, each of which represents a feasible pair of team configurations for the two teams in period t. The cost of the arcs connecting a node (kl, k2, t 1) in periOd t 1 to a node (k1', k2 t) in periOd t is given by C1, et (#t, Xt, b, r) + CE= cy (xjt1, xjt) + CE~ oj (xjt), where Zjk/~t = 1 for j = 1, 2, for t = 1, .. ., T. However, for a given node (kl, k2,ti the upper limit on either the expected patients' time in system for any patient class or the cost of modifying the team configurations is violated, i.e., either constraints (34) or constraints (35) is violated, then the cost of the incoming arcs to this node are set to M~, where M~ is a very large number. Finally, each node in 1 e. rrt = T is connected to a superficial sink node D only with zero arc cost. Figure 31 provides an example of the network representation for HCTSCP for K1 = 3, K2 = 3, and T = 2. For each team type (al, a2) represents a configuration where the number of type e personnel in the team is ae for e = 1, 2. In this figure, a path from the superficial source node to S the superficial sink node D represents a team capacity plan over the planning horizon. The HCTSCP problem finds a capacity plan with minimum cost, if the shortest path on this graph does not contain any arc with cost M~. Otherwise, the problem is infeasible. In the next section, we explain how we obtain the average time in system for each patient class. (2,4) (1,), 1(2,4), (1,1), 2 (2,4), (1,2), 1 (2,4), (1,2), 2 S ((2,4), (2,2), 1 (2,4), (2,2), 2 (2,5) (1,), O(2,5), (1,2), 1 (2,5), (1,2), 2D (2,5), (2,2), 2 (3,4), (1,2), 1 (3,4), (1,2), 2 (3,4) (2,), 1(3,4, (22), Figure 31. An illustration of the network representation for HCTSCP 3.4 Queueing Analysis In the development and analysis of the underlying queueing models, the E/T rooms, rather than health care teams, are viewed as the servers because a team may serve more than one patient simultaneously, while each E/T room can be occupied by only one patient at a time. The service of a patient is considered to begin once the patient enters a server, i.e., an E/T room, even though all the members of the associated team may be busy with other patients and the patient may have to wait in the room. To analyze the time that each patient class spends in the system, we assume that the patient arrivals in both classes are Poisson processes, and the service times for the patients are exponentially distributed with patientclass and teamtypedependent service rates. Given the service rate of each E/T room type for each patient class, we can represent such a health care system by a continuous time Markov chain (CT31C) model. We develop a DBA method to estimate the average time that the class 1 and class 2 patients spend in a health care facility for the preemptive and the nonpreemptive cases. 3.4.1 Preemptive Case: Emergency Medicine Services The evolution of the ED in a given period t of the planning horizon can be represented by a CT31C. To simplify the notation, the time index t is ignored in this section. To characterize the status of the system, we need to know not only the number of class i patients in the system (health care facility), denoted by ni for i = 1, 2, but also the number of class 2 patients treated in the type 1 E/T rooms, denoted by m. Therefore, we use a triplet (nl, n2, m) to represent the state of the system, and the associated state space is defined by S, = { (nl, n2, m) : 0 < ni < bl, O < n2 < b2, (r1 ni1 A 2n ra2 I 1(r ni1 82 a) Note that because class 1 patients have preemptive priority, if the number of class 1 patients is not less than the number of type 1 E/T rooms, there is no class 2 patient being served in the type 1 E/T rooms, i.e., m = 0 when rl < nl. The transition rate out of state (nl, n2, m) E Sp is the sum of arrival and departure rates of both patient classes, that is, Atl(ni < bi) + Xa2 82 < b2) 1 [n 1 11 (r0li 21~ [(n2 ) 72 22~~. The transitions entering state (nl, n2, m) arT Summarized in Table 31. Using the lexicographical order sequence for the states (nl, n2), the infinitesimal generator Q of the CTMC described above has the form of a nonhomogeneous quasi birthdeath (QBD) process and has the stationary probability of matrixproduct form iT,1 = jT,,_lRnl, where vector xr, represents the stationary probabilities of the states Table 31. The possible transitions enter state (n n2, m) for the ED setting Event 1) A class 1 patient arrives to the system 2) A class 1 patient arrives to the system and preempts a class 2 patient served in a type 1 E/T room 3) A class 2 patient arrives to the system, and is admitted immediately to a type 2 E/T room or joins the queue of class 2 patients 4) A class 2 patient arrives to the system, and is admitted immediately to a type 1 E/T room 5) A class1patient departure results in either freeing a type 1 E/T room or starting the service of a waiting class 1 patient 6) A class1patient departure results in starting the service of a waiting class 2 patient 7) A class2patient departure from a type 1 E/T room results in starting the service of a waiting class 2 patient 8) A class2patient departure from a type 2 E/T room results in starting the service of a waiting class 2 patient 9) A class2patient departure from a type 1 team results in freeing a type 1 server Transition probability X1P(ny AzP(ni ri) 1, n2, 1~~ > 0) 1, n82 1~~l > 0)I(n +m XA2 81, n2 2, m0~a>m XA2 81, 2 1, m 1)I(m > 0)I(n +m < rl)I(n2 m = 2a [(n + 1) A rl] lP(ni +l 1, m82 1 < b ) (ni + 1) liP(ni + 1, n2,m 1~~n bl)I(m > 0)I(ni + m = ri)I(n2 2ra m~~Pn~~ 21 1 2 2 < b2)~ 0)I(nl + m = r )I(n2 2 ~ [(n2 +2 22> 1~alaa~l 2a 21 ~i b2) (m + 1) 21P~ln 1 l82 2 ~~n b2)~ < 1l with first dimension equal to nl [75]. Although the stationary probabilities of such a process can be obtained through matrixanalytic methods, our goal is to develop fast approximation methods to obtain the average time in system for each patient class with sufficient accuracy. To estimate the stationary probabilities, we develop a DBA method with the following steps: (1) decompose the state space into three submodels, for each of which the stationary probabilities can be computed easily, and (2) combine the results of the submodels using the linking probabilities to obtain the stationary probabilities for the original CTMC and compute the average time in system for each patient class. Submodels. The state space of the CTMiC for the preemptive case can be decomposed into three submodels. Let NI~ and N2~ be random variables that denote the number of class 1 and class 2 patients in system, respectively. The first submodel determines the stationary probabilities of the number of class 1 patients in the system, NT~. To determine the stationary probabilities of the number of class 2 patients in the system, N2~, the state space S, is decomposed with respected to the first dimension by NI~ > rl and NI~ = nl for nl < rl. This decomposition procedure forms the second and third submodels, which are used to determine the stationary probabilities in regions NI~ > rl and NI~ = nl for nl < rl, respectively, i.e., the conditional probabilities P(N2~ 82 1aV > 1) and P(N2~ 82 1~V = 1) foT nl1 r1. The details of the submodels are described as follows: * Submodel 1: P(NIV = ur): The arrival and service rates of class 1 patients are independent of the value of N2~ because class 1 patients have preemptive priority over class 2 patients. Therefore, the stationary probabilities P(NIV = ur) of the original CTMiC can be determined by using an M/M/~rl/bl queueing system with arrival rate At and service rate fit. * Submodel 2: P(N2~ 82 1~V > 1): Given NI~ > rl, all type 1 E/T rooms must be occupied by class 1 patients and no class 2 patient is served in the type 1 E/T rooms. Therefore, the service rates of the class 2 patients are independent of the value of NI~, and the stationary probabilities P(N2~ 82 na1 > 1) can be determined by using an M~/M/r2/b2 queueing system with arrival rate a2 and service rate #22 * Submodel 3: P(N2~ 82 1~V = 1) foT n1 I 71I 1V r1, all type 1 E/T rooms must be occupied by class 1 patients or at least one type 1 E/T room is empty, but a waiting class 2 patient can be admitted to an empty type 1 E/T room once a class 1 patient leaves an E/T room and no class 1 patient arrives to the system. Therefore, for NV~I and one possibly available type 1 E/T room which is occupied by a class 1 patient, i.e.,ps, = (T2 82 22~~ + 2n r02 A 1r n1 olna 82 ra2 21, Where Pos is approximated by Pos, = (nlfit + Xa2lnl 1 11+ 1 2 Linking Probabilities. The unconditional probabilities P(N2~ = 2) can be represented by the conditional probabilities as follows: n1 71 P(,N2~=na P82 li 2 12 1 i 71 IP(V I 1 71 2i 82 122N 11)P~C 1 n1 ni=0 Note that the linking probabilities, P(NIV > rl) and P(NIV = n ) for nl < rl can be obtained from the results in submodel 1. If we know the stationary probabilities P(NIV = nl) and P(N2~ = 2), We can calculate the expected number of patients of each class accordingly. Then, by applying Little's formula, we can obtain the average time in the system for each class 1 and class 2 patients. 3.4.2 NonPreemptive Case: Outpatient Clinic Services The evolution of the OC in a given period t of the planning horizon can also be represented by a CT31C. As before, we again ignore the time index t to simplify the notation. The status of the system is also characterized by the triplet (n n2, m) defined for the ED. However, the state space associated with the OC setting is different from that in the ED setting and is defined by So, = {(ni, n2, m) : 0 < ni < bl, O < n2 < b2, (1 ni1 A 2n ra2 2 71 n i) Note that if all type 2 E/T rooms are occupied, i.e., n2 > 2, 811 arTTillg class 2 patient can be served in an empty type 1 E/T room, i.e., rl nl > 0. Thus, there is no state (nl, n2, m) With m less than (ri ni)t A (n2 02  The transition rate out of state (nl, n82 m E Sup is the sum of arrival and departure rates of both patient classes, that is, Atl(ni < bi) + Xa2 2 < b2) 1 [n 1(r m0~ 11 ma 21 [(2 ) 72 22.aa The types of transitions entering state (nl, n2, m) are the same as the ED setting, except that the second type of transition in Table 31 does not occur in the OC setting because the class 1 patients only have nonpreemptive priority over class 2 patients. Similar to the preemptive case, the matrix Q of the CT31C for the nonpreemptive case has the form of a nonhomogeneous QBD process, for which we estimate the stationary probabilities using the DBA method. The DBA method for the nonpreemptive case is slightly different than the one for the preemptive case and includes the following four 1! in r~ steps: (1) approximate the threedimensional CTMC by a twodimensional one; (2) decompose the state space into four submodels, for each of which the stationary probabilities can be computed easily; (3) combine the results of the submodels to obtain the stationary probabilities for the approximate twodimensional CT3 iC and then compute the expected number of patients of each class; and (4) represent the system by Class 1 Type 1 Patients I lTeam Subsystem 1 Pasins ,' TTeaem2 Subsystem 2 Figure 32. Twodimensional CTMC approximation two independent simple subsystems and use the performance measures of these simplified subsystems to adjust the results obtained in step (3). Statedimension Reduction. The dimension of state space can be reduced from three to two. We can treat the OC as a combination of two interacting subsystems as shown in Figure 32. Subsystem 1 contains the type 1 team, type 1 E/T rooms and the patients that are served in the type 1 E/T rooms. Similarly, subsystem 2 contains the type 2 team, type 2 E/T rooms and the patients that are served in the type 2 E/T rooms. We then use a twodimensional CTMiC with state (z, n2') to approximate the original threedimensional CTMC, where z is the total number of patients in subsystem 1 and n82 is the number of class 2 patients in subsystem 2. Note that z includes the number of class 1 and class 2 patients in subsystem 1, and n2' includes the number of class 2 patients served in subsystem 2 only. The state space of the twodimensional CTMC is defined by S's, = {(z, n2') : 0 < z < rl + bl, O < ~ 82~ 72 r1) + [b2 (z bl)t]I(z > r))}. Since the OC has patient classdependent service rates for type 1 team, we need to know the number of patients in each class to calculate the service rate of subsystem 1. Let Z be a random variable that denotes the total number of patients in subsystem 1, M~ a random variable that denotes the number of class 2 patients in 11 st i. h 1 and N2~ a random variable that denotes the number of class 2 patients in subsystem 2. Similarly, let Pc, be the probability that a patient in subsystem 1 belongs to class 2. We first assume that given Z = z, M~ follows a binomial distribution with parameters min(z, rl) and Pc, Then, to approximate the expected number of class 2 patients in subsystem 1 given Z = z, E [M Z = z], we need to know whether there is a queue in subsystem 1 since it affects the entries of class 2 patients to Isthi1.11 1! Therefore, we have two cases to analyze: * Case 1: For 0 < z < rl, no class 1 patient waits in the system, and the next patient entering subsystem 1 can be either a class 1 or a class 2 patient. An arriving class 2 patient can enter Isthir1 11. 1 only when all type 2 E/T rooms are occupied by patients and no other class 2 patients wait in queue, i.e., N2' r2. Thus, the total arrival rate in this case can be approximated by Ai+Xa2(V 2 r2Z < rl). Therefore, for 0 < z < rl, E[MIZ= z] can be approximated by E [M Z = z] = zP,,, (38) where Pc, = [Xa2(V 2 r2Z < ri)]/[At + Xa2(V 2 r2Z < r )]. Note that the probability P(N2~ = 2Z < rl) can be approximated by P(N2~ = 2Z < rl) from submodel 1 below. * Case 2: For rl < z < rl + bl, all type 1 E/T rooms are occupied by patients, and only class 1 patients can enter subsystem 1. Thus, the existing class 2 patients in s h ir. 11. 1 must satisfy two conditions: (1) the service time of the existing class 2 patients must be greater than the sum of the interarrival time of class 1 patients in queue, and (2) the existing class 2 patients must have entered Ithi.1. il 1 before there is a queue of class 1 patients. Therefore, for rl < z < rl + bl, we can approximate E [M Z = z] by E[MIZ = z] = max(rlPc,, TiS, + (z bi)(1 Pc )), (39) where~r Pc I[A 2Z < ri))/(At + Xa2(V 2 ra2Z < rl))]zrt. Note that under some particular parameter settings, we could have E[MIZ = z] a 0 and E[NI~Z = z] k because Pc,, O and Nz~ + M~ = Z, for example, when the value of < 21 Or Z is large enough. However, the number of class 1 patients in the OC can be bl at most. Therefore, for the state (z, n2 ) With z > bl, E[MIZ= z] should be adjusted by (z bl) (1 Pc,) as in equation (39). Submodels. After approximating the OC by a twodimensional CTEiC, we apply the DBA method to estimate the stationary probabilities of the system. The state space S',, can be decomposed with respect to the first dimension by Z < rl and Z > rl and the second dimension by N2~ > 2 and N2' r2. Then, the DBA method uses some wellknown queueing models to determine the stationary probabilities in each region, i.e., conditional probabilities P(N2~' = jZ < rl), P(Z = zN2~ > 2), P N2~' = jZ > rl), and P(Z = zN2~ I 2). The details of the submodels are described as follows: * Submodel 1: P(N2~' = jZ < ri): Given Z < ri, no class 2 patient waits in I11lir. 11. 2 because at least one type 1 E/T room is available in subsystem 1. Therefore, the value of N2~ Can Only range from 0 tO T2, Which means P(N2~' = jZ < rl) = 0 for T2 < j < b2, and the system can be approximated by an M/M/~r2r 72queut with arrival rate a2 and service rate #22 * Submodel 2: P(Z = zN2~ > 2): Given N2~ > 2, all type 1 E/T rooms must be occupied by patients in subsystem 1. Otherwise, the waiting class 2 patients would overflow to subsystem 1 and be served in a type 1 E/T room. Therefore, the value of z can only range from rl to rl + bl, which means P(Z = zN2~ > 2) = 0 for 0 < Z < rl. This system can be viewed as an M/M/1/lrl + bl queue with arrival rate At and statedependent service rate #z given by ~z = (r1 E[MIZ = z])~i 1+ E[MZ = z]#21, (310) where the value of E [M Z = z] follows from equation (39). * Submodel 3: P(N2~' = jZ > rl): Given K > rl, all type 1 E/T rooms are occupied by patients. The behavior of subsystem 2 under this condition can be modeled as an M/M/~r2/b2 queueing system. For state j with j > r2, Subsystems 1 and 2 may interact. That is, a waiting class 2 patient in subsystem 2 is admitted to a type 1 E/T room if no class 1 patient waits in 111 i.1 1!! 1, i.e., Z = rl. Therefore, the service rate of state j with j > r2 is the sum of the service rates of all type 2 servers and a possibly available type 1 server, i.e., py = r2~ 22 P,, ri, where pe, is computed from equation (310) and Pm, is the probability that a type 1 server is available for a waiting class 2 patient, and can be approximated by Ponz = P(Z = rlN2 72 ra)i, 2 Xllr, 2 a X1 * Submodel 4: P(Z = zN2~ I 2): Similar to Submodel 3, we model the behavior of the subsystem 1 as an M/M/~r /rl + bl queueing system. For state z with z < rl, the subsystems 1 and 2 may interact. That is, an arriving class 2 patient overflow to subsystem 1 and receives service in an empty type 1 E/T room if all type 2 E/T rooms are occupied by patients and no class 2 patient waits in subsystem 2, i.e., N2' r2. Thus, for z < rl, the arrival rate of state z is the sum of arrival rates of class 1 and possible class 2 patients, i.e., Az = X1 + Po,,12, Where Pos, is the probability that an arriving class 2 patient receives service in a type 1 E/T room, i.e., P(N2~ = Z < rl) from submodel 1. The service rate pz can be computed by z, = (min(z, ri) E[M1Z= z]) ~11 + E[MIZ= z]#21, 31 where the value of E[MIZ = z] is computed from equation (38) for 0 < z < rl, and from equation (39) for rl < z < rl + bl. Linking Probabilities. The unconditional probabilities P(Z = z) and P(N2' j can be represented by the conditional probabilities as follows: P(Z = z) = P(Z = zN2~ 72 2~PI~ I 2) + P(Z = zN2~ 72 2~PI~ > 2) and P(N2 ~ 2 )= (V' = jZ < r )P(Z < ri) + P(N2~' = jZ > ri)P(Z > r ). Note that the linking probabilities, P(N2~ I 2) and P(Z < rl), can further be represented by the conditional probabilities as follows: P(N2~ 72 2a = 72'I Z < r )P(Z < ri) + P(N2~ I 2Z > r )P(Z > ri) and P(Z < rl) = P(Z < rlV' N T72(V~ 2 r2) + P(Z < rlV' N 2~PI~ 2 r2  If we substitute P(Z > r ) = 1 P(Z < rl) and P(N2~ 72 2a = 72~'I) above, we can solve the resulting equations to obtain: P(N2~ I 2)= and 1 P(N2~ > Z > r )P(Z < rl' N 2~ P(Z < rl) = P(Z < rllN2 72 2~PI~ 72 ~ Therefore, after computing the conditional probabilities in all the four submodels, the unconditional probabilities, P(Z = z) and P(N2~' = j), and the expected number of patients in subsystem 1 and 2, E[Z] and E[N2 ', TOSpectively, can be computed. Then, E [NI] and E [N2], can be computed using the following set of equations: rl+bl E[M] = E[MZ= z]P[Z:= z], (312) z=0 E [N~] = max {E [Z] E [M], O }, and (313) E [N2] = E [N2'] + E [M]. (314) Again, by applying Little's formula, we obtain the average time in system for each class 1 and class 2 patients. Bounds. To improve the performance of the proposed DBA method for the nonpreemptive case, we use some simple queueing systems to obtain bounds on E [Z], E [NI], and E [N2], and then adjust the results from equations (312)(314). *Expected total number of patients in subsystem 1 (E[Z]): The lower and upper bounds on E[Z] are obtained by considering two M/M/~rllrl + bl queues with statedependent service rates from equation (311) with arrival rates At and At + XA2 2 respectively. Let f2 be the fraction of class 2 patients served in the type 1 E/T rooms, which is approximated by f2 (I~ 2 r2, Z < rl) + P(N2' r2, Z = rl) 2 X1 + X2 SP(N2~ = 2Z < rl)P(Z < rl) + P(N2~ = 2Z > rl)P(Z = rl) 2.(:315) X2 X1 Note that the first term in equation (:315) represents the probability that an arriving class 2 patient finds all type 2 E/T rooms occupied and overflows to Isthi.(h il 1 immediately. The second term represents the probability that an arriving class 2 patient waits in subsystem 2 until a type 1 E/T room becomes available. Let E[NIV] be the upper bound of E [NI]. If E [Z] is greater than E [NI], then E [NI] and E [Af] in equations (:312) and (:313) are decreased proportionally, i.e., E[NIV] E[NIV] E [NIV]ne = E [Ni]previous and E [Af]nw = E[Af],revious E [Z] E [Z] Otherwise, i.e., if E [Z] is less than the upper bound of E [NI], E [NI] and E [Af] in equations (:312) and (:313) are increased proportionally. * Expected number of class 1 patients in the system (E[NI]~): The lower and upper bounds on E[NIV] are obtained by considering two Af/Af/rl/bl queueing systems with service rates fit and with arrival rates At and At + X2 2, Tespectively. * Expected number of class 2 patients in the system (E[N2]I) The lower and upper bounds on E [N2] are obtained by considering two multiserver queueing systems. Let E[Y] be the total number of patients in an Af/Af/rl + T2/b2 queueing system with arrival rate At + X2 and service rate p., where rl + T2 < b2. The service rate y is assumed to be pL = (AT Ag /(A/1 2,wer fi h fetv arrival rate of class i patients and is approximated by bl+rl A~ = I(1 P(Z =/\L D)) andl A2 A2(1 2(V = b2 . z=bl The resulting E[Y] is a lower bound on the total number of patients in the OC because class 1 patients can he served in the type 2 E/T room here. Thus, the lower bound on E [N2] can he computed by E [Y] E [NI]. Last, the upper bound on E [N2~ is obtained by considering an Af/Af/T2/b2 queueing system with arrival rate X2 and service rate 22*. 3.5 Computational Study In this section, we present results from our computational study, where we first investigate the efficiency and the accuracy of the DBA method. We then use the DBA method in conjunction with the HCTSCP formulation to test the efficiency of our capacity planning model in making longterm personnel planning decisions. 3.5.1 Computational Performance of the DBA Method To assess the computational performance of the DBA method, we compare the approximate results obtained by the DBA method with the exact results obtained by solving the steady state equations using Gaussian Elimination (GE). In preliminary experiments we observe that GE may require significant computational effort as the size of the model increases. Therefore, we consider an adaptation of Gaussian Elimination (AGE) proposed by Thorson [106]. AGE algorithm avoids unnecessary row operations by considering the fact that Q is a banded matrix, containing a large number of zero elements. The DBA, GE, and AGE methods are implemented using C++ programming language, and the numerical results reported are obtained using a personal computer with a 3.0 GHz Pentium IV processor and 1 GB RAM memory. Our parameter choices for our computational study are based on the data collected from a participating ED. In the base case, there are eight and two rooms allocated to type 1 and type 2 teams, respectively, i.e., (rl, r2) = (8,2); service rates for class 1 and class 2 patients in type 1 E/T room are 0.25 and 0.80 patients/hour, respectively, i.e., ( II, 2~1 =(0.25, 0.80); service rate for class 2 patients in type 2 E/T room are 0.75 patients/hour, i.e., 422 = 0.75; and the maximum number of patients from each class allowed in system are 20 and 20, respectively, i.e., (bl, b2) = (20,20). We note that in practice, an ED is required by law to admit all the patients that request emergency medicine services. Therefore, essentially, the waiting room capacity is infinite, and no arriving patient is denied of entry to the system because of lack of waiting room capacity. However, when there is not enough service capacity, then an arriving patient can be diverted to a sister hospital. Therefore, for modeling purposes, we include a limit on the number of patients of each type in the system. In our study, we consider three experimental factors including the E/T room allocation, service capacity, and system utilization. In the first and second experiments, we test the impact of E/T room allocation and service rate on the accuracy of DBA, respectively. We consider rl e {2, 4, 8} and T2 E {2, 4, 8} for E/T room allocation in the first experiment and (411, #21) E { (0.20, 0.70), (0.25, 0.80), (0.30, 0.90) } and 22~ E {0.65, 0.75, 0.85} for service rates in the second experiment. Let p denote the system utilization. For each scenario, we use pe { 0.6, 0.7, 0.8, 0.9} to generate four instances with different pairs of patient arrival rates, i.e., (X1, X2), using At rlfirp and X2 = r2 22  In Table 32, we report the size of the CT31C as well as the CPU time (in seconds) required to obtain the solution using the DBA, AGE, and GE methods. In the preemptive case, the size of the CT31C model grows if rl or T2 illCTreSeS. In addition, an increase in rl has a more significant effect than that in T2. Similar behavior can be observed in the nonpreemptive case, however, the impact of increasing rl on problem size in nonpreemptive case is more significant than that in the preemptive case. Furthermore, under the same E/T room allocation, the problem size of nonpreemptive case is at least twice larger than that of preemptive case for the tested scenarios in Table 32. Our results show that the time to obtain the approximate results using the DBA method is negligible and AGE is considerably more effective than GE in obtaining the exact solution. Table 32. Computational requirement of the DBA, AGE, and GE methods Preemptive NonPreemptive Ti T2 State Count DBA AGE GE State Count DBA AGE GE 2 2 447 0.0 0.0 0.5 2707 0.0 0.3 10.8 2 4 453 0.0 0.0 0.6 2713 0.0 0.3 10.9 2 8 4635 0.0 0.0 0.6 2725 0.0 0.3 11.2 4 2 461 0.0 0.0 0.6 4225 0.0 0.7 36.6 4 4 481 0.0 0.0 0.7 4245 0.0 0.7 37.9 4 8 521 0.0 0.0 0.9 4285 0.0 0.7 39.8 8 2 513 0.0 0.0 0.8 6609 0.0 1.9 116.1 8 2 585 0.0 0.0 1.2 6681 0.0 2.0 129.2 8 2 729 0.0 0.0 2.3 63825 0.0 2.1 161.0 Unit of running time: Second In Tables :3:3 and :34, we report the percentage error associated with the expected time in system for the two patient classes obtained by the DBA method (when compared to the exact solution obtained by the AGE method) in the preemptive and nonpreemptive case, respectively. Table :3:3 shows clearly that the average time in system for class 1 patients can he correctly computed by the DBA method because submodel 1 in Section :3.4.1 captures the exact behavior of class 1 patients in the original CT~iC. For class 2 patients, the first experiment shows that for the instances with the same levels of rl and p, the absolute percentage error (APE) tends to decrease aS T2 increases. For example, for the instances with rl = 2 and p = 0.6, APE decreases aS T2 illrerOSes. In addition, for the same levels of rl/r2 and p, APE tends to decrease as rl or T2 increases as system utilization is low, such as the instances with rl/T2=1 and p=0.6, APE decreases as rl (or r2) increases. In other words, for a given setting of rl or rl/T2 and p, DBA performs better in the problems with larger T2. In the second experiment, 9 scenarios of service capacity are tested. For the same level of ( 11, 21), APE tends to increase as 022 illrerOSes for the instances with p=0.6, which is opposite to the results of the instances with p=0.9, where APE tends to decrease as 022 increases. Last, both experiments show that DBA tends to underestimate class 2 patients' expected time in the system when the system utilization is low, i.e., p = 0.6, while overestimate class 2 patients' expected time in the system when the system utilization is high, i.e., p = 0.9. Table :34 shows the results for nonpreemptive case. In the first experiment, we observe that for the instances with the same levels of rl and p, the APE for class 2 patients tends to decrease aS T2 illrerOSes, which is the same as the preemptive case. In the second experiment, for the same levels of ( II, 21) and p, APE tends to increase as 022 increases for both class 1 and class 2 patients. In addition, DBA tends to underestimate class 1 patients' expected time in system while it tends to overestimate class 2 patients' expected time in system. For class 1 patients, APE is less than 5 percent for all tested instances, i.e., DBA yields a more reliable estimate for class 1 patients' time in system. h h crc~ d" vvv ~cC~ o;~d 6~6~6~ t~u3~ ~t~b; cu~~ vvv cCb~G9 o;~b; b~b~cC 6~6~6~ G9~0 t~cj, cucu~ vvv b~u3u3 o;~b; u3u36~ 000 moddcm o h 000 6 do c 000 lyp IIv 6~6~ m dC`9C~' ~t~b; vvv ~u3~ b;dt: hhh 6~6~6~ b~~c~ b;b;~ vvv C13CUG9 LljodCd h h h ;6~6~ 6; C~)b~ b~~crj d vvv C13G9u3 d~~c~c~~ 000 000~6 000rj 000 hhh 6~6~6~ cutcu crjb;cd vvv b~G9~ V V VIV V CUC~u3C13t~ crj crj cjlcj cj co oo oc h~lCl b~ C~ICU b~ C~ICU b~ C~ ~6~6~ 6: ooo "dd o;iiii u5u5ii o;cuuj 6~6~ ~ioo cid cuc~ d;d; 6~6~ iiO r'cu iii0 6~6~6~ c~3ii~ ujcdcd S~6~6~ iiii~ cucici cu r i;lci crj i; c~3 CU ii iii ii C~ OOC\C\ IC cuc usu ON6 ocu cuc cucr O~6 c0o Ocd O I I C ooi a~o CU~~u5~CU ed o; o;lo; d i; c99 ?? C i I o NTT~CICuTTUT T 60 r oO ; c~o ICU 6m6 OOc cu u cu ii usooo icM Obus . : u ii ssli s dddddddddsuusu INCIm m l s l e 0 C 0 C 0 3.5.2 Computational Performance of the HCTSCP Model To illustrate the effectiveness of the HCTSCP model used in conjunction with the DBA method in making longterm personnel planning decisions, we compare the capacity plans generated by the HCTSCP model using the approximate and exact average time in system for different patient classes obtained by the DBA and AGE methods, respectively. We also study the sensitivity of the results obtained by the HCTSCP model to several problem parameters. We consider the personnel planning problem over a threeyear planning horizon where the unit planning period corresponds to a quarter of a year, i.e., T = 12. We assume that, for all t, the initial service rates for class 1 and class 2 patients are (0110, #210, 220o (0.25, 0.80, 0.75) patient/hour, the allowable maximum numbers of patients in system are (bl, b2) = (20,20) patients, the E/T room allocation is (rl, T2) = (8,2), and the upper bound on the amount of time that a patient spends in the system are (caz, ag~) = (4.75, 4.00) hours. We consider a case where there are two types of personnel included in each team, and there are six and four feasible configurations for type 1 and type 2 teams, respectively. Table 35 shows the number of different types of personnel with different skill sets included in each configuration and the corresponding service capacity as well as operating cost for each team. Without loss of generality, we assume that patient waiting costs increase linearly with patients' time in system, unit delay cost for each patient class (UPit, UP~t) arT Set to ($400,$100) /hour per patient, and the funds that can be allocated to change the service capacity of the teams in the facility 7t= $17,000 for all t. Finally, we assume that personnel hiring or termination costs are zero. An examination of emergency medicine practices show that patient arrivals to the ED exhibit seasonality. We generate the total quarterly arrival rate of patients using a seasonally adjusted trend line, represented by a function of the form Xt = 6mod(t,4)U( 0 + bt) where be is the quarterly seasonality factor for season i (where we have the following estimates for the seasonality factors by = 0.8, 62 = 1.0, 63 = 1.2, and 64 = 1.0), a is Table 35. Team configurations Type 1 Team Type 2 Team kl al a2 ~11 21COSt k2 81 82 2 COSt 1 2 4 0.23 0.75 56,000 1 1 1 0.60 11,500 2 2 5 0.25 0.80 63,000 2 1 2 0.75 16,500 3 2 6 0.28 0.85 70,000 3 2 2 1.00 23,000 4 3 4 0.35 1.00 70,000 4 2 3 1.10 28,000 5 3 5 0.40 1.10 77,000 6 3 6 0.45 1.20 84,000 Unit of '. patients/hour. Unit of cost: $/quarter a uniformly distributed random number (where we have u ~ U[0.8, 1.2]), Ao = 2, b = 0.04, and t = 1, .. ,12. Note that we implicitly assume that the patient demand increases linearly by 2 percent every quarter. We assume that the fraction of class 1 patient is fit = 0.85 for all t, i.e., Xlt = flt~t and X~t ( lt Xt. We note that our parameter choices are mainly according to the characteristics of the data collected from the ED, which is represented by the preemptive case. In order to eliminate the effects that may be due to a specific health care facility, we use the same set of parameters for the nonpreemptive case also. In our study, we considered three experimental factors including the fraction of class 1 patients, fit, the unit patient treatment cost for each patient class, (UPit, UP~t), and the maximum allowable average time in system for each patient class, (cx, c82). Each parameter is tested at three levels as listed in Table 36. For each experimental setting, we generated 25 random instances of the patient arrival streams over the planning horizon. We solved all the instances for each of the settings by using the HCTSCP model with the AGE method and with the DBA method to compare the performance of the two methods. Table 36. Parameter settings Parameters Level 1 Level 2 Level 3 Experiment 1 flt(~ ) 80 85 90 Experiment 2 U~it ($/hour/patient) 200 400 800 UP~t ($/hour/patient) 50 100 200 Experiment 3 atl (hours) 4.50 4.75 5.00 a2~ (hours) 3.50 4.00 4.50 As explained in Section :3.3, we build a network to represent the HCTSCP problem. To compute the patient delay cost for each are, we obtain average time in system for each patient class approximately (exactly) using the DBA (AGE) method. We note that we do not use the GE method in this experiment, since the AGE method is shown to be considerably more efficient in Section :3.5.1. After we construct the network, we find the shortest path using Dijkstra's algorithm [2]. The results show that the network for the 12period HCTSCP problem with six and four feasible configurations for type 1 and type 2 teams, respectively, contains 290 nodes and 6,384 arcs. The average time required to obtain the optimal capacity plan is 11.0 seconds for preemptive case and 549.8 seconds for nonpreemptive case if we use the AGE method in building the network for an instance of the HCTSCP problem. In contrast, the DBA method requires 0.03 seconds, on average, for both the preemptive and nonpreemptive cases. Therefore, using the DBA method in building the network for an instance of the HCTSCP problems is considerably more efficient than using the AGE method. In order to measure the accuracy of the DBA method, we compare the team capacity plans generated by the two approaches based on a ;.0 .1.;<./.;, indexr. Specifically, we count the number of periods where team configurations chosen are the same in both approaches and then divide this counting by 12 for each care team type to determine the value of the similarity index. The results of our comparison for each experimental factors are summarized in Tables :37, :38, and :39. Cells range from 0 percent (absolutely different) to 100 percent (perfectly similar). For instance, if a cell has a value of 75 percent, then among 12 periods, the HCTSCP model with DBA gives the same results in 8 of them as the HSCTSCP model with GE. In Table :37, we observe that the HCTSCP model with DBA works very well in the preemptive case. However, in the nonpreemptive case, as the fraction of class 1 patients increases, the performance of the HCTSCP model with DBA deteriorates as it overestimates the required service capacity of type 2 team by choosing the team C11 oa oh cvj " ~~100 3 ~ld or o o oa oh~ ri ~~100 b 3 ~ld F 01 . Cr3 a O ~lcr3 Ln ~,II` cu 011, oa o~o; or ~~103 CI~ a," 3ad ho 01 . oa o~o; ri ~~103 CI~ > a," orh o 3ad b a Ln ~,103 u a,~ ~to a~8 iD1 13 I cY3r ~cd oooo L 000 ~000 a co 0000 000 ~000 L 000 000 ~000 a co Ln 000 0000 ad coo rr ~cd oooo cY30 ~oo oo dd oo o;o; oo dd oo rr o;o; 000 coooo 0000 00 ~b Ln oooo configuration with a higher service rate. This can he attributed to our earlier observation that the DBA method overestiniates the average time in system for class 2 patients. Table 38 shows that the unit patient delay cost does not impact the accuracy of the HCTSCP model in the preeniptive case. But in the nonpreeniptive case, its accuracy in determining the team configuration of type 2 team goes down, as unit class 1 patient delay cost or unit class 2 patient delay cost decreases, and it tends to overestimate the the required service capacity of the type 2 team by 1 level. Table 39 shows that the nmaxiniun allowable average time in system does not materially impact the accuracy of the HCTSCP model in either the preeniptive case or the nonpreeniptive one. In suninary, the HCTSCP model with DBA is efficient and accurate in solving the capacity planning problems, particularly in the preeniptive case, e.g., the ED application. For the nonpreeniptive case, e.g., the OC application, its accuracy in type 2 care team requirements is not as precise. 3.6 Concluding Remarks and Future Research Directions In this paper, motivated by the emergency medicine services at a coninunity hospital and specialty services at an outpatient clinic, we examined the health care team service capacity planning problem. These health care systems provide services to two 1!! ri ~ patient classes using two types of health care teams and two types of E/T rooms. While class 1 patients are served by the type 1 team in the dedicated E/T rooms only, class 2 patients can he served hv either the type 1 or the type 2 team in the E/T rooms dedicated to the corresponding team. However, the type 1 team delivers service to a class 2 patient only if there are no class 1 patients waiting in the system and all type 2 E/T rooms are occupied by other class 2 patients. The service capacity of each team is measured by the service rates of the dedicated E/T rooms for each patient class. Although this setting is similar to some other studies in the literature, the distinguishing characteristic of our work is that we have patientclass and teamtypedependent service rates. In our work, we first considered the case where the class 1 patients can preenipt class 2 patients served in type 1 E/T rooms, which is typical in emergency rooms in hospitals. We also analyzed the nonpreemptive case, common in outpatient clinic settings. We developed queueing models for both cases, and developed approximation procedures to estimate the average time that each patient class spends in the system. Through an extensive computational study, we illustrated that our DBA method provides the performance measures of interest efficiently with sufficient accuracy. Using the results of the queueing an~ lli we then developed a nonlinear binary integer programming model to determine the minimal cost capacity plan of health care teams that a health care facility should employ over a planning horizon to deliver service for the patients while ensuring that the average time that each patient class spends in the system does not exceed certain values. Our computational study showed that our approximation approach provides sufficiently accurate results that can be used in practice to make longterm health care team service capacity planning decisions. We note that in our queueing analysis, we assumed exponentially distributed interarrival and service times to preserve analytical tractability. Extending our work to consider general arrival and service processes is a potential area for future research. Moreover, in our study we analyzed systems where patients are categorized into two classes. Although this classification is widely used in the health care industry, some clinics and hospitals further classify each of the classes into two or more subclasses [100]. The presence of multiple subclasses in each patient class magnifies the problem size rapidly and cannot be solved through general numerical methods (e.g., the GE or the AGE method). Therefore, developing approximation procedures for such settings would be of theoretical and practical interest. In our work, we focused on the longterm capacity planning of the health care teams, treating the E/T rooms as the servers and assuming that the service rates of the rooms for different patient classes are given, and room allocation is fixed over the planning horizon. For more detailed analysis, the set of tasks required for the diagnosis, monitoring, and treatment of a patient can be modeled using a queueing network, and the service rates of the E/T rooms under different room allocations and team configurations can be further investigated. CHAPTER 4 HOSPITAL BED ALLOCATION PROBLEM 4.1 Introduction In this chapter, we introduce the hospital bed allocation (HBA) problem. The HBA problem is an extension of the AHBCP problem in C'!. Ilter 2, which is concerned with determining the optimal .I__oegate bed capacity plan over a finite planning horizon. After the .I_ gate bed capacity is specified, the next step involved is concerned with the allocation of .l__oegate bed capacity among different medical care units (j!CI~s) (e.g., neurosurgery, oncology, pediatrics, etc.). Ineffective allocation of existing bed capacity among different medical service units can lead to service quality problems for the patients along with operational and/or financial inefficiencies for the hospitals. An arriving patient may be declined or placed on hold by an MCIT, if there is no bed available to accommodate the patient in the unit. In this case, the patient may be subject to same health risks due to the necessity to find an alternative health care provider or wait until a bed becomes available. From the hospital's perspective, the potential revenue is either lost or deferred, which may lead to some financial inefficiencies. Similarly, an arriving patient may be accepted for treatment but can he accommodated in another MCIT (e.g., an arriving obstetrics patient can he boarded in neurosurgery). In this case, the patient may be subject to same unnecessary health risks due to the necessity to be boarded together with patients with more different, possibly more serious, health conditions. From the hospital's perspective, the potential revenue is collected but the operational costs may increase, as the patient may be boarded in an MCIT where the service resources are more expensive (e.g., neurosurgery nurses may have additional qualifications than obstetrics nurses and the medical equipment in a neurosurgery department are typically more expensive than the ones in obstetrics). To effectively utilize existing bed capacity, hospital administration could choose from among a number of alternative planning strategies to to find an allocation of existing .I_ egate bed capacity among different MCUs. In particular, there are four practical planning strategies, including * the expected bed occupancy is balanced across entire hospital, * the expected net profit of the hospital is maximized, * the occurrence of bed shortages is minimized, or * the number of patients rejected is minimized. In this work, we focus on the first strategy and develop a mathematical programming formulation to address this problem. We also develop effective solution approaches to obtain high quality solutions particularly for largesized, realistic test instances. The remainder of this chapter is organized as follows. In Section 4.2, we review the related literature. Section 4.3 presents mathematical programming formulation for HBA. We develop three heuristic solution approaches in Section 4.4. In Section 4.5, we present results from our computational study that evaluates the computational performance of the approaches developed in Section 4.4. Section 4.6 includes a discussion of the results and II_  r future research directions. 4.2 Literature Review Most health care managers apply relatively simple approaches, such as the use of target occupancy level with average length of stay, to forecast bed capacity required for a hospital or an MCU. Yet, the failure to adequately consider the uncertainties associated with patient arrivals and time needed to treat patients by using such simple approaches may result in bed capacity configurations where a large portion of patients may have to be turned .l.li [46]. To take the stochastic nature of health care systems into account, researchers utilize queuing and simulation models in determine the appropriate bed capacity configuration. The application of queueing theory allows for the evaluation of the expected (longrun) performance measure of a system by solving the associated set of flow balance equations. Mackay and Lee [80] evaluate the choice of models for forecasting bed capacity and II__ r to use compartmental flow model, which models patient flow through a hospital as flow through a sequence of compartments. Patients with short length of stay may leave the system after visiting the first compartment, otherwise, patients move to the next compartment until their length of stay are reached. The benefit of using compartmental flow model is that it can capture the variation in bed occupancy without using a sophisticated method. Gorunescu et al. [43] model a department of geriatric medicine as an Af/Af/c/K queueing system to investigate the interrelationships between admission rates, length of stay, number of allocated beds, and probability that an arriving patient is denied admission. K~ao and Tung [63] present an approach for allocating beds to care units in a hospital to minimize the expected patient overflow, i.e., the rate at which patients are denied admission due to inavailability of bed capacity. They model each service as an Af/G/oc queueing system and use normal approximation in computing number of patient overflows. The bed allocation problem is solved in two stages. The first stage distributes the 1 in 4 Gly of beds such that no gross imbalances in bed utilization among all care units are observed and a prespecified fraction of patients can stay in the units designated for the unit. The second stage uses marginal analysis to allocate the remaining beds to minimize the expected total patient overflow. Discreteevent simulation is useful for the analysis of systems with complex behavior, including health care systems. Discreteevent simulation has been widely applied in health care services [58] to study the interrelationships between admission rates, hospital occupancy, and several different policies for allocating beds to AICUs. Harrison et al. [52] construct a simulation model where patients' stay in hospitals are classified into three stages, which represent different phases of care provided. The output obtained from the model matches the mean and the variability associated with actual bed occupancy data. The model is used to identify daily occupancy distributions, study tradeoffs between overflow and bed capacity levels, and investigate the effects of various changes. Akkerman and K~nip [6] use Markov chain approach to specify the number of beds needed for two hospital wards, and then utilize discreteeventsimulation to obtain detailed information on expected bed occupancy and patient rejection levels. Masterson et al. [84] use discreteevent simulation to investigate the interrelationships between bed occupancy, average number of patient deferred, and different bed allocation and operating policies. Harper [51] develops a simulation model for the planning and management of hospital beds, operating theaters, and workforce needs. The model captures the complexity of health care systems by incorporating the variability for each patient group such as monthly, daily, and hourly demand as well as the distributions of length of stay and operation times. K~im et al. [68] analyze an intensive care unit with 14 beds and develop a simulation model to evaluate different bedreservation schemes to reduce the number of cancelled surgeries. Some papers consider the hierarchical relation between care units. For example, after a mothertobe delivers her child in the labor and delivery unit, she should be moved to the postpartum unit for recovery. When the capacity downstream is insufficient, patients are forced to stay at the current care units with typically more expensive equipment blocking the capacity at these upstream care units. To take the interactions among care units in a hospital into account, Cochran and Bharti [30] first apply queueing network methodology (without blocking) to find a balanced bed allocation, which is obtained through trialanderror work. Then, they use simulation analysis to estimate the blocking behavior and patient sojourn times. Galviio et al. [39, 40] apply a threelevel hierarchical, capacitated model to determine the capacity required in perinatal health care facilities, which is categorized into three levels: basic units, maternity homes, and neonatal clinics where intensive care unit for babies is available. In our work, we utilize a different approach by integrating results from queueing theory into an optimization framework. Specifically, we model each MCU in a hospital as an M/M/~c/c queueing system to estimate the probability of rejection when there are c beds in the unit. We then develop an optimization model to allocate the .I__oregate bed capacity across different MCUs. The purpose of this work is to develop efficient solution approaches to solve this problem. 4.3 Problem Formulation We now begin the mathematical formulation with the objective of balancing bed occupancy throughout the hospital. We consider a hospital with D MCUs and B beds available. For each service there are lower and upper limits on the number of beds allocated, denoted by 14 and ui, respectively. In addition, there is a lower bound on the bed occupancy for each MCU, denoted by y. To analyze the bed occupancy of each MCU, denoted by pi, we assume that the patient arrivals of each MCU are Poisson processes and the lengths of stay at each MCU are exponentially distributed. We assume that an arriving patient to MCU i is rejected, if all beds designated for the MCU i are occupied. Let pi denote the probability of rejecting an arriving patient of MCU i, and xi be the number of beds allocated to MCU i. Let p be the bed occupancy of the entire hospital, As be the patient arrival rate and 1/ps be the average lengths of stay at MCU i. Let xi be the decision variables of HBA problem, i.e., the number of bed allocated at MCU i. We then formulate the HBA problem as a nonlinear integer programming formulation as follows: min p p (41) i= 1 subject to x4=B (42) i= 1 xi! m! m=o (1 pi) As pi = Vs (44) p = (45) i= 1 le < xi < ai Vi46 < < i 1 Vi47 xi E Z* Vi(48) The objective function (41) minimizes the total deviation of the bed occupancy for each of the MCUs from the overall average bed occupancy for the hospital. Constraint (42) limits the total number of beds allocated among the MCUs to the number of total beds available in the hospital. Constraint set (43) represents the probability of rejecting patients arriving to service i as a function of arrival rate As, service rate ps and bed capacity xi for service i. Constraint set (44) represents the bed occupancy of the service i as a function of effective arrival rate (1 pi)As, service rate ps and bed capacity xi for each service i. Constraint (45) specifies the overall average bed occupancy for the hospital. Constraint sets (46) and (47) impose the lower and upper bounds on the bed capacity and the bed occupancy for each service i, respectively. Finally, constraint (48) ensures that the decision variables are nonnegative integers. HBA is a difficult nonlinear binary integer programming problem with nonlinear constraints. Note that constraint set (43) involves the factorial function on xi and the summration of the termn~I fr~omn m = 1 up to m11 = xsl w~hic~h increases the c~omplexuity of the problem significantly. Since decision variables xi's assume discrete values, we can introduce the splitting variable yij, where yij takes value of one if xi = bij and zero otherwise, where bij E {li, 14 + 1, ..., as 1 ui}. Let pij and pij represent the bed occupancy and probability of rejecting an arriving patient of MCU i, respectively, when bed capacity of MCU i equals bi. As a result, we can obtain an equivalent linear binary integer programming problem that can be stated as follows: D ug 14 minl boys;V (4 9) i= 1 j= 1 subject to D u 1 b oysZ,i = B (410) i= 1 j= 1 pi (Ag/ps)bqiC (i in" p = bii! m!= V i, (411) pij = V j(412) D ug 14 p = (413) i= 1 j= 1 < I y =1 i(415) j= 1 sij > Pij P i (416) sij > P payV (417) yij E {0, 1} i (418) be, {ls14+1...as 1us} i~j(419) We have four sets of additional constraints. Constraint set (415) ensures that only one splitting variable takes the value of one. Constraint sets (416) and (417) compute the deviation of the bed occupancy for each service i from the overall average bed occupancy for the hospital. Constraint set (418) ensures that the decision variables take binary values. Finally, constraint set (419) provides information of the discrete options of 4.4 Solution Algorithms The HBA problem with variable splitting formulation can be solved by commercial MILP solvers, however, it takes time to obtain the optimal solutions of realsize problems. This section describes three solution approaches we develop for the HBA problem, which includes genetic algorithm (GA), greedy randomized adaptive search procedure (GRASP) and a hybridization of GA and GRASP. 4.4.1 Genetic Algorithm GA constructs a population of solutions and generates new generations of solutions mimicking the behavior of population genetics [41, 53]. In particular, two members of the current population of solutions are chosen randomly and used to generate new offspring which are then retained for the next generation of solutions if they qualify. Figure 41 depicts the pseudocode of the GA. A preprocessing step ensures that a test instance is feasible by verifying the validity of the following inequalities: le i< B (420) i= 1 us > B(421) i= 1 p(14) > 7 V (422) where p(14) represents the bed occupancy with bed capacity 14 of service i. procedure GeneticAlgorithm(instance) 1Read and preprocess the input data; 2 Solution* < 0; 3 Population < GeneratePopulations (PopulationSize); 4 NumGenNotImprove < 0; 5 for m = 1 to MaxNumGenerations do 6 Children < ProduceOffspring(Population, NumGenNotImprove); 7 Population < UpdatePopulation(Population, Children); 8 Solution* < Up dateSolution( Solution*, Children, NumGenNot Improve); 9 end for end GeneticAlgforithm. Figure 41. Pseudocode of the genetic algorithm GA starts by generating a set of distinct feasible solutions to form an initial population of solutions. These solutions are generated through an occupancydriven approach, and then adjusted to feasible by a randomrectified approach (see Figure 42). The occupancydriven approach first treats each MCU as an M/M/cl queueing system, and then allocates a number of beds to the MCU such that the bed occupancy is close to y (see Figure 43). As it can be observed from constraint set (43), specifying the probability of rejecting a patient from an MCU that is modeled as an M/M/~c/c queueing system requires the knowledge of the number of beds allocated to that particular MCU. Since our objective is to generate some solutions for the initial population of GA, rather procedure GeneratePopulations (PopulationSize) 1 Population < 0); 2 while NumlnSet (Populaiton) < PopulationSize do 3 x < OccupancyDrivenAllocation(A, p, y); 4 x < RandomRectified( x, 1, u, y); 5 if x Sf Population then 6 Population < Population U x; 7 end if 8 end while 9 return (Population); end GeneratePopulations. Figure 42. Pseudocode of the population generating procedure than spending time to find the corresponding bed capacity for an M/M/~c/c queueing system, we model the MCU as an M/M/cl queueing system to initialize the bed capacity. procedure OccupancyDrivenAllocation(A, p, y) 1 x < 0; 2 for i = 1 to D do 4 end for 5 return (x); e nd O ccup ancyD rivenAllo cation. Figure 43. Pseudocode of occupancydriven allocation The solutions generated from the occupancydriven approach may not satisfy constraint sets (42), (46) and (47), i.e., constraint sets associated with the total number of beds available, lower and upper bounds and bed occupancy (of an M/M/~c/c queueing system). We use a randomrectified procedure to finetune the initial bed allocation. Figure 44 shows the pseudocode of the randomrectified procedure. We first revise the bed capacity of each MCU such that the constraints (46) and (47) are satisfied. Then, when the total number of beds allocated is greater (less) than B, an MCU is chosen randomly and its bed capacity is decreased (increased) by one, if this operation does not violate constraint (46). The step of choosing an MCU randomly and adjusting its capacity accordingly is repeated until the total number of beds allocated is equal to the overall bed capacity available, B. procedure RandomRectified( x, 1, u, y) 1 for i = 1 to D do 2 while p(xi) < y do 3 xi < xi 1; 4 end while 5 xi < min~xi, ui}; 6 xi < max~xi,14}>; 7 end for 8 while NumBedAllocated(x) < B do 9 i < RandomlySelectInteger(D); 10 xi min~xi +1i, u ; 11 end while 12 while NumBedAllocated(x) > B do 13 i < RandomlySelectInteger(D); 14 xi < max~x { x, 1i,1 } 15 end while 16 return (x); end RandomRectified. Figure 44. Pseudocode of randomrectified procedure After the initial population is formed, GA starts to produce offspring. Two types of genetic operators are considered in this work; one is simple singlepoint crossover, and the other is mutation. In simple singlepoint crossover (see Figure 45), we first randomly select two parents from the population, and then swap bed capacity between the randomly selected [D/2] MCUs. That is, if the MCU i is selected, the bed capacity of MCU i of the first parent's chromosome is swapped with the bed capacity of MCU i of the second parent's chromosome. The resulting solutions are the offspring solutions produced by the selected parent solutions. Figure 46 shows an example with four MCUs and 100 beds, where the second and the third MCUs are chosen to swap their bed capacity. Note that the crossover may produced an offspring which does not have a chromosome specifying a feasible solution. In this case, the randomrectified approach is emploi a to adjust the bed capacity allocation. The second genetic operator, mutation, is invoked if the current best solution is not improved throughout the evolution of a prespecified number of consecutive generations. The mutation employs the occupancydriven approach with the average occupancy procedure Crossover(Population) Sx < RandomlySelectElement (Population); 2 y RandomlySelectElement (Population); 3 while y= x do 4 y < RandomlySelectElement (Population); 5 end while 6 SwapSet < 0); 7 while NumlnSet(SwapSet) / [D/2] do 8 i < RandomlySelectInteger(D); 9 if i ( SwapSet then 10 SwapSet < SwapSet U i; 11 end if 12 end while 13 for i = 1 to D do 14 if i E SwapSet then 15 temp < xi; 16 xi < ys~; 17 ye~ < temp; 18 end if 19 end for 20 x < RandomRectified(x); 21 y < RandomRectified(y); 22 if f(x) < fly) then 23 return (x); 24 else 25 return (y); 26 end if end Crossover. Figure 45. Pseudocode of crossover procedure (10, 20, 30, 40) (10, 16, 24, 40) (38, 16, 24, 22) (38, 20, 30, 22) Figure 46. Example of crossover of the current best solution to produce an offspring. Again, this mutated offspring is adjusted through the randomrectified procedure to ensure feasibility. Then, the value of NumGenNotImprove is set to zero. If the offspring produced from either crossover or mutation procedure does not exist in population and is better than the current worst solution in population, then the current procedure Mutation(x*) 1 p* p (x*); 2 x OccupancyDrivenAllocation(A, p, p*); 3 x RandomRectified( x, 1, u, y); 4 NumGenNotImprove< 0; 5 return (x); end Mutation. Figure 47. Pseudocode of the mutation procedure worst solution in the population is replaced with the offspring to form a new generation with the other existing solutions. Otherwise, i.e., either the offspring exists in population already or the offspring is not better than the current worst solution in the population, this offspring is ignored and the next offspring is produced using the procedure described above. Note that the current best solution is also updated if the new offspring has a better objective function value than that of the current best. 4.4.2 Greedy Randomized Adaptive Search Procedure Greedy randomized adaptive search procedure is a multistart approach [36] that is widely used for combinatorial optimization problems. Each GRASP iteration consists of two phases: construction and local search. Figure 48 depicts the pseudocode for GRASP. The construction phase creates a feasible solution, whose neighborhood is explored by the local search phase to find a locally optimal solution. The algorithm stops after a prespecified number of iterations is executed, which is denoted by MaxNumIterations in Figure 48. procedure GRASP(instance) 1 Read and preprocess the input data; 2 Solution* < 0; 3 for n = 1 to MaxNumIterations do 4 Solution < GreedyRandomizedConstruction (a); 5 Solution < LocalSearch (Solution); 6 Solution* < Up dateSolution( Solution, Solution*); 7 end for end GRASP. Figure 48. Pseudocode of GRASP The construction phase builds a feasible solution in a greedy manner. First, each MCU is allocated an initial bed capacity at its lower bound to form an incomplete solution. Then, one bed is added to an MCU if this capacity expansion does not destroy the upper bound constraint and the increment on the objective value is at an acceptable level. The pseudocode of the construction procedure is illustrated in Figure 49, where the ej in line 22 represents a zero vector except the jth element equals one. Lines 15 through 20 in Figure 49 build a restricted candidate list (RCL), which records the set of MCUs for which adding one more bed to the MCU does not violate the feasibility and has the potential to improve the objective function value at an acceptable level. The threshold of increment on the objective function value is controlled by the parameter as {0(, 1} on line 17 in Figure 49. An MCU is included in RCL, if adding one more bed to the unit is feasible, and the increment on the objective function value is not greater than 6min + a~(6max 6min), where 6min and 6max represent the minimum and maximum increments on objective function value after incorporating one more bed to current solution, respectively. Note that the lower the a~ is, the greedier the procedure is. A bed is added to an MCU selected randomly from RCL, until the total number of beds allocated equals B. During the local search phase, the neighborhood of the feasible solution created in the construction phase is fully investigated to find the local optimum. A neighbor solution is produced by swapping one bed from one MCU to another, if this swap does not destroy the feasibility of the solution. The pseudocode of the local search procedure is illustrated in Figure 410. Here, we adopt the bestimproving strategy that is we evaluate all feasible neighbors and choose the neighbor that improves the objective function value the most. 4.4.3 Hybridization of GA & GRASP We develop a hybridization of GA and GA. The pseudocode for the hybrid approach (HA) is given in Figure 411. On each iteration of the HA, a set of elite solutions is generated, in contrast to a single initial solution used by GRASP. Then, an offspring is procedures GreedyRandomizedConstruction(a~) 1 x < 0; 2 for i = 1 to D do 3 xi = li; 4 end for 5 while NumBedAllocated(x)/ B do 6 6min t +oo; 8 for i = 1 to D do 9 if xi + 1 < ui then 10 si < f (x+ei) f (x); 11 6min <min {6min, be }; 12 6maz < max {6max,6 i); 13 end if 14 end for 15 RCL = 0); 16 for i = 1 to D do 17 if xi + 1 < ui and be < 6min + a~(6max 6min) then 18 RCL < RCL U i; 19 end if 20 end for 21 j < RandomlySelectElement (RCL); 22 x < x + ej; 23 end while 24 return (x); end GreedyRandomizedConstruction. Figure 49. Pseudocode of greedy randomized construction procedure produced using the crossover procedure of GA and improved by the local search procedure of GRASP. The previous two steps are repeated, until a prespecified number of iterations are completed. The characteristic of the HA is that on each iteration the threshold a~ for the greedy randomized construction phase is updated by a decreasing function p(u), i.e., a~ decreases as the number of iterations, n, increases. In this way, the HA can test more than one setting of a~ and reduce the probability of converging to a local optimum solution prematurely. To form an elite set of solutions, we first use the GreedyRandomizedConstruction procedure of GRASP to generate NumGRASP distinct solutions, and then choose the best ESize of them, where NumGRASP > ESize. The pseudocode is presented in Figure 412. procedure LocalSearch(x) 1 x*<x; 2 f*< f (x); 3 for i = 1 to D do 4 for j = 1 to D do 5 if i / j, xi 1 > 14, xi + 1 I ui and p(xi + 1) > y then 6 if f (xei + ej) < f then 7 x*<xei + ej; 8 f*< f (xei + ej); 9 end if 10 end if 11 end for 12 end for 13 return (x*); end LocalSearch. Figure 410. Pseudocode of local search procedure procedure Hybrid(instance) 1Read and preprocess the input data; 2 Solution* < 0; 3 for n = 1 to MaxNumIterations do 4 ~ <p (u); 5 EliteSet < GenerateEliteSet(a~, Solution*); 6 for m = 1 to MaxNumGenerations do 7 Solution < Crossover(EliteSet); 8 Solution < LocalSearch( Solution); 9 Solution*< UpdateSolution(Solution, Solution*); 10 end for 11 end for end Hybrid. Figure 411. Pseudocode of HA 4.5 Computational Study In this section, we present results from our computational study, where we investigate the computational efficiency of the proposed solution approaches. Specifically, we measure the efficiency of the approach using the CPU time needed to obtain the solution and relative error in the objective function value, given by IA f* procedure GenerateEliteSet(a~, Solution*) 1 EliteSet < 0); 2 for j = 1 to NumGRASP do 3 x < G reedy Randomiz ed Construct ion ( a); 4 while xe EliteSet do 5 x < GreedyRandomizedConstruction(a~); 6; end while 7 y < the worst solution in EliteSet; 8 if f(x) < fly) then 9 EliteSet < EliteSet \ y U x; 10 end if 11 Solution*< UpdateSolution(x, Solution*); 12 end for 13 return (EliteSet); end GenerateEliteSet. Figure 412. Pseudocode of elite set generation procedure where fA denotes the objective function value obtained by using approach A, where AE {GA, GRASP, HA}. In our study, we consider three settings each with different instance sizes to evaluate the impact of the size of the instance on the performance of the proposed solution approaches. The problem size is varied with the number of total beds available in the hospital, B, and number of MCUs in the hospital, D. For the total number of beds and the number MCUs available in the hospital, we consider three levels that correspond to small, medium, and large sized hospitals. In particular, we consider hospitals with 750, 1,000 or 1,250 beds and 30, 40 or 50 MCUs. For each setting, we generate 30 random instances. Each random instance is obtained by generating parameters that correspond to patient arrival and service rates along with lower and upper bounds on the number of beds available in each MCU. Specifically, each parameter is obtained by using the formula: (mean value)+u, where u is drawn from the distribution U [0.4, 1.6]. The mean values of the random parameters are listed as follows: 1. Mean arrival rate (Ag): 10 persons/per unit time for each MCU i; 2. Mean service rate (ps): 0.5 persons/per unit time for each MCU i; 3. Mean lower bound (14): 10 beds for each MCU i; and 4. Mean upper bound (ui): 80 beds for each MCU i. Last, the minimum bed occupancy, y, is 'TI' for all MCU all problems. The experiment is implemented on a workstation with two Pentium 4 3.2 GHz processor and 6 GB of memory. To obtain a nearoptimal solution as a comparison basis, we use CPLEX to obtain the optimal solutions for the test instances. For each test instance, CPLEX is stopped if the relative stopping tolerance of 0.01 is satisfied, or CPU time of 3,600 seconds is used. Table 41 reports the results obtained by CPLEX. As we expect, the number of instances for which CPLEX can determine and verify the optimal solution within one hour of CPU time decreases as the problem size increases. Moreover, for the instances with small problem size, CPLEX takes more than 1,800 seconds on average to solve one instance. 4.5.1 Summary of Results Obtained by GA GA is executed for 200D generations and performs a mutation when best solution is not improved for D generations, where D is the number of MCUs in consideration. To evaluate the impact of the population size, we consider three levels of D, 2D and 3D for each setting. Table 42 reports the results of GA, where the information is classified into two lI;ir The first by. ;r distinguishes the instances with different sizes, and the second specifies the parameter setting of GA, including number of generations and population size. Note that if the relative error of an instance is less than zero, it is replaced by zero when computing the average relative error of the corresponding instance. In general, GA performs quite well in terms of average relative error and CPU time. For any of the 90 instances, GA spends less than 2 seconds to find a solution and the average relative error is less than "' Furthermore, for more than one third of the test instances, GA finds solutions better than CPLEX. For example, for the fourth largesized instance, GA finds a solution with an objective function value which is about 5.1' lower than that of the solution found by CPLEX. Table 41. Nearoptimal solutions obtained by using CPLEX Problem size (D, B) Instance 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 27 28 29 30 No. of instances solved by CPLEX Small (30, 750) Objective Time 0.1964 0.6132 0.2574 > 0.2869 0.4828 > 0.4588 0.2817 0.2484 0.1351 0.4558 0.3447 0.4320 0.2872 > 0.4589 0.5883 > 0.2848 0.2751 > 0.1942 > 0.3408 > 0.2600 0.3262 > 0.4457 > 0.1813 0.3991 > 0.6148 > 0.3832 > 0.8989 0.3392 0.1377 0.4448 Medium (40, 1000) Objective Time 0.2691 0.4122 > 0.6417 > 0.4141 > 0.7636 > 0.3180 > 0.5691 > 0.4530 > 0.3737 > 0.3347 > 0.4972 > 0.6381 > 0.7216 > 0.6496 > 0.8077 > 0.5183 > 0.2022 0.6919 0.4640 > 0.6155 > 0.5205 0.4363 > 0.1789 > 0.3602 > 0.7231 > 0.7193 > 0.5233 > 0.6959 > 0.4833 Large (50, 1250) Objective Time 0.5298 > 1.0745 > 0.5083 > 0.7238 > 1.0353 > 0.2228 > 0.5841 > 0.4073 > 0.6679 > 1.1034 > 0.5721 > 0.6960 > 0.4622 > 0.7431 > 0.4568 > 0.9320 > 0.5609 > 0.5702 > 0.7131 > 0.6928 > 1.0242 > 0.6448 > 0.7533 > 0.7442 > 0. lII 10 > 0.5765 > 0.7700 > 0.4138 > 1.0881 > (sec.) 37 4 3,600 1,972 3,600 21 1,369 709 2,111 110 1,277 47 3,600 4 3,600 375 3,600 3,600 3,600 8 3,600 3,600 8 3,600 3,600 3,600 12 730 1,770 13 (sec.) 318 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 543 29 3,600 3,600 1,727 3,600 3,600 3,600 3,600 3,600 3,600 3,600 577 (sec.) 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 3,600 The parameter of population size does not appear to have significant impacts on relative errors and CPU times. For example, for mediumsized test instances, when the population size grows from 30 to 90, the average relative error changes from 1.15' to 1.55' and the average CPU time slightly increases from 1.01 seconds to 1.05 seconds. Moreover, about N I' of the 30 instances obtain the relative errors at the same level with respect to three different sizes of population. E 9999qq o~~m~~~m~~~m~~~mov a 0 a0 :S O ccomeomo0omoaemom~a c al ac Oi r do a c o a o oa o v c c 0 0 o~omo om~o co~ocacc~ooo coooo OMO dddd0000+0 dHNddNHMMO ON S ~10~~~00~~~00~~O~~~0000000000000000 O m c~~oovoocodovo~~ocodmoacmocccaacis 0 00000000000000000~000000000000000 OO I cj o o .50 o i 30~ i~ Ed ~m~~~~a~o~m~~~~~a~~~87O 4.5.2 Summary of Results Obtained by GRASP GRASP is executed for 20D starts (iterations). The threshold parameter, a~, of construction phase is examined at 0.25, 0.30 and 0.35 three levels. Table 43 di pE.ls the results of GRASP. As is obvious to see that GRASP takes CPU times longer than GA to obtain solutions. GRASP spends 3.1 seconds in average to obtain a solution of the problem with small size, in contrast to GA's 1.6 seconds with respect to the problem with large size. In general, the accuracy of GRASP and GA does not appear significantly different, if we compare the best results from each of them. For example, in the problem with medium size, the relative errors of GRASP (with a~ = 0.25) are 1.;:T' and 8.2 !' in average and maximum, respectively, and GA (with population size = 40) are 1.15' and 9.45' respectively. Furthermore, among 30 instance GRASP (with a~ = 0.25) finds results better than CPLEX for 9 instances, compared to 11 instances of GA (with population size =40). Table 43 presents the tendency that the lower a~ is, the lower the average relative error is. However, a low value of a~ has the drawback that it may lead the solution to a local optimal solution and result in a large error. For example, the instance 20 in problem with medium size has relative error 8.2 !' when a~ is 0.25. In contrast, the instance has less relative error, 3.'7;' when a~ is increased to 0.35. In GA and GRASP, we do observe that some instances have relative error greater than >' such as the instance 14 of the problem with medium size in Table 42, and the instance 20 of the problem with medium size in Table 43. To reduce the errors of those instances, we combines these two algorithms to develop a hybridization. 4.5.3 Summary of Results Obtained by HA HA is run for [D/10] iterations, in which 200D generations are produced. In the beginning of each iteration, we generate 2D distinct solutions and pick the best D of these solutions to form an elite set. The added feature of HA is that a~ decreases as the number of iterations increases. To evaluate the effects of a~ on the proposed approach, we consider ~Om 0+0 emmmowb CHHOmm~ombamobabe ~O~~O ~ aboice monarieva~swom01mm o onew e a a a weemmmemer~l~~ 00 i 00 QU MU O 00+0 00 0 o coco somove a 0 on aaon, ~ 00 CI0 O t0umOt ~o~~ovo~ocatoma ca~ a oo as ddddmdd~d~dmdiddmddid a~ oD .cooCI .0. co .cob co~ 0000 ~ ~ ~ oO COM CI00 0+ 0 t ON O N @@0~0000000000 0 0000000~00O~~000 amonomooaaoacemo~ace .moom egg em ~d~ddddddddddtddddddd~ddddd oma st Ommeme~mememememeammeme oOam miea lom o m000000a000[000000~ 000 ttOM UNHmc a comiococo a amonoaco mea a~ io 0 c o coo oc co co co v O be LO 00000 ONNOOOttM~~O 00[00+0000 + 0000 000000 o00000000~00O 0 O ,~a E"X o mm~~clW E~ac~' d~ E~E~ a~e~~F, ~E ~7 ~ O 'Ci 0" ~ drdd o l a m o o o a o to o cu o , three updatingf functions of a~ as follows: Constant: pl(u) = 0.3, Linear: p2 (n) = 1.0 0.7 ,and Nonlinear: p3(u) = 1.0 0.7* where NV equals [D/10] and is the total number of iterations. To provide a benchmark, we include the constant function pl(u) in our experiment, where a~ is set to 0.3 on each iteration a for n = 1,2,..., [D/10]. Using function p2 8), We linearly decrease a~ to 0.3 as the number of iterations increase. Using function p3 8), Wt decrease a~ to 0.3 in a nonlinear fashion over the iterations. Note that a~ values generated by function p3 8) 1S Ilo larger than those generated by function p2(u) for each n as shown in Figure 413. 1.0 0.8 0.6 1 ******* pl(n) 0.4 p2(n) 0.2 p3(n) 0.0 1 2 3 4 5 Figure 413. The updating functions of a~ The results are shown in Table 44. In each test instance, the first column reports the relative errors and CPU times obtained by using the updating function pl(u), and so on. For the mediumsized test instances, the average (maximum) relative error decreases from 1 >'.~ (9.3 !'.) to 0.;:".'.~ (4.11 J.'.), when p2 8) is used instead of pl(u) The function p3 8 , which has smaller a~ than p2(u) at each iteration n, does not appear to make HA perform better than the HA with p2 8). FOT eXample, the average relative error for largesized instances increases from 0.35'.~ to 0.4!'.~ if the p2 8) is replaced by p3(n). 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This also supports the viewpoint mentioned in GRASP, i.e., with a low value of a~, the approach may converge to a local optimum prematurely. HA outperforms both GA and GRASP in terms of solution quality. The relative error is less than 0.5' in average and 5' in maximum for all problems whether p2(u) or p3 8) is used. Furthermore, HA (with p2(u) or p3(u)) finds better solutions than CPLEX for about half of the instances, compared to GA or GRASP's one third. As to the computational requirement, HA spends less than 80 seconds to obtain a solution for the largesized instances, which is larger than either GA or GRASP, but is still at an acceptable level from a practical perspective. From these, we can conclude that HA is an effective and efficient solution approach for the HBA problem. 4.6 Concluding Remarks and Future Research Directions In this chapter, we focused on allocating the available .I_ egate hospital bed capacity among MCUs to balance the bed occupancies among different units. To take the uncertainties associated with health care systems into account, closedform results from queuing theory were incorporated into an optimization framework, which resulted in large scale nonlinear integer programming formulations for the HBA problem. To efficiently solve the problem, we proposed three solution approaches. Our computational study showed that GA and GRASP were very efficient, both of which solved the problem within 30 seconds with average and maximum errors less than ;:' and 10I' ., respectively. To decrease the maximum relative errors observed in GA or GRASP, we proposed a hybridization of GA and GRASP, denoted by HA. HA outperformed either GA or GRASP in terms of accuracy, whose relative error is <0.5' and <5 in average and in maximum, respectively, by spending <80 seconds in terms of CPU time. In summary, HA is an efficient approach that finds good quality solutions for the HBA problem. In our work, each MCU is modeled as an M/M/~c/c queueing system. Future research can consider different queueing system configurations with general arrival and service processes or queueing networks. The design of the queueing network can model the hierarchical relation between subunits in each 1\CU, i.e., patients flow through units in a specific order. We note that the mathematical expressions of stationary probabilities of a queueing network would be more complicated than that of an Af/Af/c/c queueing system. Specifically, the expressions of stationary probabilities of a queueing network have a product form of capacity allocated in each station, which correspond to the the decision variables of the HBA problem. Another practically relevant research direction is concerned with the modeling of the blocking behavior between subunits in each 1\CU. Note that the blocking in a health care system is different from that in a manufacturing system, where the work of a blocked part can not he started before entering a designated station. However, the recovery process of a patient who is blocked from entering the bed in a downstream unit does not stop. For example, a mothertobe is typically allocated to a bed in the labor and delivery unit first, and moved to a bed in postpartum unit to recover from delivery. It may happen that the mother spends her entire recovery time in the labor and delivery unit and is discharged from there directly. Because she fully recovers in the labor and delivery unit before a bed becomes available in the postpartum unit. CHAPTER 5 EMERGENCY ROOM SERVICES FACILITY LOCATION AND CAPACITY PLANNING, 5.1 Introduction This chapter introduces the emergency room services facility location and capacity planning (ER SFLCP) problem. In traditional facility location models, facility location and demand allocation decisions are made based on the objectives of minimizing the total number of facilities opened or minimizing the total or weighted) distance traveled. This approach ignores the fact that some demand may not he satisfied due to a shortage of capacity or system congestion as the system operates in realtime. In a health care service facility, when the emergency room (ER) is full or all intensive care beds are occupied, hospitals send out divert status. When a hospital is on divert status, incoming patients might he sent to hospitals which are farther away or kept at the hospitals where they currently are that may not able to provide adequate service. To a critical patient, the consequence of divert status can he the difference between life and death. The purpose of this work is to construct a facility location model, which simultaneously determines the number of facilities opened and their respective locations as well as the capacity levels of the facilities so that the probability that all servers in a facility are busy does not exceed a predetermined level. In other words, we want to locate ER services on a network and determine their respective capacity levels such that the probability of diverting patients is not larger than a particular threshold. The remainder of this chapter is organized as follows. In Section 5.2, we review the related literature. Section 5.3 presents mathematical programming formulation for ERSFLCP. Section 5.4 details the Lagfrangfian relaxation algorithm that we propose for the ERSFLCP problem. In Section 5.5, we present results from our computational study that evaluates the computational performance of the Lagrangian relaxation algorithm. Section 5.6 includes a discussion of the results and II__ ; future research directions. 5.2 Literature Review Traditional facility location models (such as the set covering model, the Pmedian model, or the Pcenter model) focus on determining the location of facilities and the allocation of demand to facilities with the objective of either minimizing the number of facilities opened, minimizing the sum of fixed facility location and/or transportation costs, or maximizing the demand covered. In practice, however, a customer may choose to go to a facility different from the one identified by the optimization model for a number of reasons, such as the designated facility experiences a temporary shutdown, is associated with long queues and wait times. There is a rich body of literature that develop robust facility location models under uncertainty to hedge the randomness on costs, demands, and travel times utilizing the robust or stochastic optimization approach. Snyder [103] presents a detailed review of on stochastic and robust facility location models. In the area of health care application, Beraldi et al. [12] investigate the problem of characterizing the optimal locations of emergency medical service sites and numbers of emergency vehicles required for each site. They consider the problem formulation in a stochastic optimization setting, where the ability to cover the random requests of emergency service at demand points is restricted by a set of probabilistic constraints. Specifically, the probabilistic constraints ensure that the probability that the number of vehicles located at a facility can cover the random service request is greater than a prescribed probability value. Some papers consider the hierarchical relation between facilities in a health care setting. For example, K~oizumi et al. [70] classify mental health care system into levels of extended acute hospitals, residential facilities and supported housing, and patients flow through the levels according to their health conditions. Because of the hierarchical structure of health care service, the capacity requirements of units in a higher level of hierarchy usually correlate with the units in a lower level. Galviio et al. [39, 40] formulate a hierarchical locationallocation model to determine the capacity required in perinatal health care facilities, which are categorized into three levels: basic units, maternity homes and neonatal clinics. They develop a mixed integer linear programming model which aims to determine the optimal locations of facilities for each level and allocations of motherstobe to these locations for needed type of service. Another branch of literature that is related to our work develops facility location models which require backup (multiple) coverage for each demand point. Snyder and Daskin [104] consider a Pmedianbased model that minimizes total costs associated with a locationallocation plan and the expected failure cost. As each demand point is assigned primary and backup facilities, the expected failure cost is quantified by the additional transportation cost incurred to cover the demand by the backup facility. The problem is formulated as a 01 integer programming formulation and solved by the Lagrangian relaxation algorithm. Jia et al. [56] present a detailed review of traditional facility location models and propose a general facility location model suited for largescale emergencies. In their model, a demand point is considered to be covered if a prespecified number of facilities are assigned. To address the issue of service quality, some papers incorporate queuing systems into facility location models to consider the randomness on availability of servers and focus on reducing the demand lost due to the shortage of capacity or system congestion. Marianov and ReVelle [81] formulate a maximal availability location model, which uses a M/M/~c/c queueing system to model the server availability of a demand point. The model wants to locate a set of ambulances such that the demand covered is maximized, where a demand point is considered to be covered, if the probability that there is an ambulance nearby and available is greater than a threshold. They show how to transform the nonlinear queueing expression to an equivalent linear one, and solve the problem by using a commercial solver (LINDO). Berman et al. [13] model each facility as an M/M/1/la queueing system, where a is the maximum number of customers allowed in the facility, and consider a facility location problem with the upper bounds on the amount of demand lost due to insufficient coverage and system congestion. Berman et al. [14] also investigate the facility location problem under the objective of maximizing captured demand. They, again, model each facility as an M/M/1/la queueing system and assume that a customer is lost if the closest facility and all other facilities that he/or she can reach are full. Both papers ([13, 14]) obtain the solutions, i.e., the location of the facilities, through heuristic approaches. We note that the capacities of facilities to be opened has an impact on both the total number and the locations of facilities, particularly when the congestion associated with the potential facilities is taken into account. In what follows, we model each facility as an M/M/~c/c queueing system, where c is number of servers in the facility, which designates the capacity of the facility. The goal of our model is to identify locations of facilities and specify the capacity levels of the facilities simultaneously. 5.3 Problem Formulation We consider a network G = (N, A), where N = {1, 2, ... N} is the set of nodes, and A denotes the set of arcs. We assume that at each node i eN the occurrence of patients needing ER services is Poisson distributed with rate us, and all patients at a node need to be directed to one open ER service facility. We consider that an ER service facility can be opened at any node j E N, and at most p facilities can be opened for providing the ER services to patients, which is similar to most of the facility location models. Let dij be the distance (measured in time units) between nodes i and j for i, jE N, and d be the coverage range of an ER. The allocation parameter aij takes the value of one if dij < d, zero otherwise. Let t represents the monetary value for travel time for a critical patient. Let fj be the fixed cost of opening a facility at node j e N, and Cyk, be the operating cost of the facility j with capacity level k E K, where K = {1, 2,..., K} is the set of capacity levels of an opened ER. In this problem, the capacity is measured as number of beds in an ER. That is, if an ER is to be opened at capacity level k E K, then there are mk beds in the ER. We assume that the lengths of stay of a patient in the ER at node j are exponentially distributed with rate py, and each ER is modeled as an M/M/~c/c queueing system. In other words, when all beds in an ER are occupied, patients are diverted (rejected) to other ERs. Let y be the upper bound of probability of diverting a patient from an ER, As be the patient arrival rate to an ER service facility at node j, and zy be the bed capacity of the ER service facility at node j. Note that zy = mk, if the facility at location j is chosen to be opened at capacity level k. We have four sets of decision variables. The first set is patient allocation variable xij, which takes the value of one if node i is served by the ER service facility at location j, zero otherwise. The second set is ER location and capacity allocation variable yjk, Which takes the value of one if an ER service facility is placed at node j and operated at capacity level k, zero otherwise. The last two sets are As and zy, which are determined once the values of xij and yjk, are aSSigned. The ERSFLCP problem can be formulated as a nonlinear integer programming formulation as follows: min tindijxij + ) ( fj + cyk)#jk (51) i= 1 j= 1 j= 1 k= 1 subject to ax = 1 Vi(52) j=1 xis < yjk Vi, j(53) yjk < p(54) j=1 k=1 yjk <1 Vj(55) i= 1 Kj(7 K Klj i)(8 k= 1 k= 1 xij, yjk E {0, 1} ,j (59) zj E Z* V (510) where xr(Ay, py, zy) is the probability of diverting patients from an ER service facility at location j. The formulas for xr(Ay, py, zy) of an M/M/~c/c queueing system can be found in any queueing book (e.g., Gross and Harris, 1998). Specifically, xr(Ay, py, zy) can be represented as a function of arrival rate Ay, service rate py, and bed capacity zy, that is x(A, (,zy =~~ )zy! q! (511)11 The objective function (51) minimizes the value of time of ER patients and the costs of opening and operating ER service facilities. Constraint (52) stipulates that each demand node must be covered by one facility. Constraint (53) restricts that demand nodes can only be assigned to opened facilities. Constraint (54) imposes the upper bound on the number of facilities opened. Constraint (55) states that an opened facility must be associated with a single capacity level. Constraints (56) and (57) obtain the arrival rates and capacity levels for all the facilities. Constraint (58) imposes the upper bound on the probability of all beds being occupied for an opened facility. Finally, constraints (59) and (510) ensure that decision variables are binary and nonnegative integers, respectively. Note that the factorial terms in equation (511) make the ERSFLCP problem intractable. To overcome this problem, we replace the constraint (58) by constraint (512), As <... ., +My(1 yj) V k(512) where '.. is the largest value which satisfies inequality (513) and equation (514). 3, mk m q 5 7 (513) Sk* 4 q=0 My = easy(514) i= 1 As a result, we transform the previous nonlinear integer programming formulation to a linear binary integer programming formulation as follows: min" ~ingdaymy + ii(j fy + yk)# (515) i=1 j=1 j=1 k=1 subject to a x, 1 Vi(516) j= 1 xi < CYjkV (5 17) yjk < p(518) j=1 k=1 yjk, < V (519) i= 1 xij, yjk E {0, 1} ,j (521) 5.4 Solution Approach The ERSFLCP problem can be solved by commercial MILP solvers, however, it takes time to obtain the optimal solutions of problems with large size or tight constraints, for example, large values of NV or small values of p. Here, we develop a Lagfrangfian relaxation (LR) approach to obtain solutions to the ERSFLCP problem. 5.4.1 Lagrangian Relaxation Approach The general idea of Lagrangian relaxation is to move hard constraints into the objective and penalize the objective if the constraints are violated. Figure 51 presents the general procedure we use to obtain a solution to the ERSFLCP problem using a Lagfrangfian relaxation approach. In Figure 51, a is the iteration counter. Also, UB* and LB* denote the incumbent upper and lower bounds on the objective function value of Lagrangian dual problem, respectively. Similarly, UB" and LB" denote the upper and lower bounds on the objective function value of Lagrangian dual problem at iteration n, respectively. u is the counter of number of iterations that UB* does not improved. Last, nmax and umax are upper bounds of n and u. In the following subsections, we discuss the elements of the iterative approach. pro ced ure Lagrangi anRelaxation() 1 n < 1; a < 1; UB* < oo; LB* < 0; 2 Initialize Lagfrangfian multipliers; 3 while n < nmax and a < umax do 4 Formulate and solve the Lagfrangfian problem to obtain LB"; 5 if LB" > LB* then 6 LB* < LB"; 7 end if 8 Obtain UB" using problem specific approach; 9 if UB" < UB* then 10 UB* < UB"; 11 u < 0; 12 else 13 a< a+ 1; 14 end if 15 Revise Lagfrangfian multipliers using subgfradient optimization; 16 n< a+ 1; 17 end while end LagfrangfianRelaxation. Figure 51. Pseudocode of the Lagfrangfian relaxation 5.4.2 Lower Bound To obtain the lower bound of ERSFLCP, we relax the constraint sets (517) and (520) with Lagrange multipliers a~ and P, respectively, where a~ and P are matrices with sizes NVx NV and NVx K, respectively. The relaxation yields the following Lagrangian problem (ERSFLCPLR): i= 1 j= 1k (fy ~ ~ .I + Cy as 'i j)yjk pjkn~ i (522) j=1 k=1 i j=1 k=1 subject to asxi = 1 Vi(523) j=1 yjk < p(524) j=1 k=1 yjk <1 Vj(525) xij, yjk E {, 1} ,j (526) Note that the Lagfrangfian problem can be separated into subproblems LX and LY, where subproblem LX contains variables xij, and the subproblem LY contains variables yjk, aS follows: (LX) min am i= 1 j= 1 subject to aixij = 1 V j=1 xi E {0, 1} i (LY) min ikj j=1 k=1 subject to j=1 k=1 where da inn'day + nmy + ne, CE jkji and cjk jtci Cjk n, s 1 sy i '~i pj. Given a set of a~ and p, both subproblems LX and LY are easy to solve. For subproblem LX, for each i eN the decision variable xij, is set to one if day, < dij for all j E N. Similarly, subproblem LY can be solved according to each variable yjk S contribution to the objective function, while for each j eN at most one yjk, can be set to one for all k E K, and at most p of the yjk S can be set to one for all j EN and k E K. Note that if yjk = 1, then cjk < 0 must hold. 5.4.3 Upper Bound In the beginning of the algorithm, we first generate an upper bound by the heuristic that will introduced later. Also, at each LR iteration, the heuristic is applied to improve the lower bound solution from the Lagfrangfian subproblem, i.e., subproblems LX and LY, to become a feasible solution. Figure 52 depicts the pseudocode of the heuristic. The heuristic starts by resetting the infeasible solution (X, Y) according to the strategy selected randomly from following: 1. For each i, if xij > 0, then CE= yjk ; n 2. f Eyjk= 0 thenl xsy = 0 for a~ll i t N. The first strategy resets facility variables, yjk, based on demand allocation, xij, determined from subproblem LX. The facilities are ordered according to their patient arrival rates, i.e., As = Ci nixij, in nonincreasing ordern, and then the first p facilities are set to open at capacity level one, i.e., yjl = 1. The second strategy resets demand allocation variables, xij, according to solution yjk, form subproblem LY. That is, if a facility j is not opened, then the demand nodes allocated to facility j are reset to not covered, i.e., the corresponding variable xij's are reset to 0 and reallocate them to other facilities using the next procedure. Line 2 resets the capacity level k of the opened facilities such that the facilities is opened at appropriate capacity level k, i.e., Ci nixij < '.... For an opened facility j, if the demand allocated exceeds its largest capacity, i.e., Ci nixij > ~jK, then an allocated demand node i is selected randomly and its xij is reset to 0, until Ci nixij < '.. holds. Next, the while loop in line 3 in Figure 52 ensures that all demand nodes are covered and the number of facilities opened is less than p. The pseudocode of covering all demand nodes is presented in Figure 53. We randomly select a node a which has not covered by procedure GenerateFeasibleSolution(X,Y) 1 ResetSolution(X,Y); 2 Reset capacity level; 3 while CiC xij < N or Cj Ck yjk > p do 4 if CiC xij < NV then 5 CoverDemand(X,Y); 6; end if 7 if Cj Ck yjk > p then 8 CloseFacility(X ,Y); 9 end if 10 end while 11 return (X,Y); end GenerateFeasibleSolution. Figure 52. Pseudocode of the feasible solution generation any facility, and generate a set OpenF which contains the facilities meeting the following criteria: 1. The facility j is opened, i.e., Ck yjk, > 0; 2. The facility j can covered the demand node n, i.e., any = 1; and 3. After allocating node a to the facility j, the sum of arrival rates of the allocated demand node does not exceed facility j's maximum capacity, i.e., Ci nixij +n, < < jK If the set OpenF is empty, then a set NVotOpenF is generated. The set NVotOpenF includes the facilities which are not opened, i.e., Ck yjk = 0, and the node u is within their coverage range, i.e., any = 1 for all j E N. Then, in the line 8 in Figure 53 we open a facility je N otOpenF to cover node u. There are many strategies that we can apply to choose which facility to open, such as, open the facility je N otOpenF which is the closest one to node n, the one with the largest capacity, or the one with the smallest fix cost. Here we apply the three strategies and find the corresponding facilities for each of them. If these strategies yield different facility options, we randomly select one of them to open. Once all demand nodes are covered by the opened facilities, the capacity level of each opened facility is reset to appropriate level k E K (line 14 in Figure 53) such that So far, we have ensured that all demand nodes are covered by facilities which are opened, and the demand allocation does not exceed the maximum capacity of the opened procedure CoverDemand(X,Y) 1 while Ci C xi < NV do 2 a < RandomlySelectElement (UncoveredNode); 3 Generate set OpenedF = {j : Ck yjk > 0, any = 1, Ci nixij + n, < < jK~j E 6 4 if OpenF / 0 then 5 j < the closest facility j E OpenF; 6i else 7 Generate NVotOpenedF = {j : Ck yjk = 0, any = 1, jE N}; 8 select j from the set NVotOpenF; 9 NVotOpenF < NVotOpenF \ j; 10 OpenF < NVotOpenF U j; 11 end if 12 xij < 1; 13 end while 14 Reset capacity level; 15 return (X,Y); end CoverDemand; Figure 53. Pseudocode of covering all demand nodes facilities. The next thing to do is to inspect whether the number of facilities opened is no larger than p. If the number of facilities opened is less than or equal to p, then a feasible solution is generated. Otherwise, one of the opened facilities is selected randomly and closed, until the number of facilities opened is no larger than p. In addition, the associated demand nodes are reset to not covered, i.e., reset xij = 0, where facility j is chosen to close. procedure CloseFacility(X,Y) 1Generate set OpenedF = {j : Ck yjk > 0, jE N}; 2 while Cj Ck yjk > p do 3 j < RandomlySelectElement (OpenF); 4 yjk < 0, for all k E K; 5 for i = 1 to NV do 6 xij < 0; 7 end for 8 OpenF < OpenF \ j; 9 end while 10 Reset X based on Y; 11 return (X,Y); end CloseFacility; Figure 54. Pseudocode of closing facilities The procedures of CoverDemand and CloseFacility are repeated, until a feasible solution is generated. 5.4.4 Lagrangian Multipliers Due to the modern software and hardware developments, the linear programming problems can be efficiently solved by any commercial LP solvers. To take advantage of the existing tools, we use the dual information of the linear form of the ERSFLCP problem, denoted by ERSFLCPr, to initialize the Lagrangian multipliers, rather than starting the algorithms from scratch as traditional approaches. The ERSFLCP, problem is generated by replacing the constraint sets (517) and (520) by 0 < xij < 1 and 0 < yjk I i, TOSpectively. Then, CPLEX is used to solve the ERSFLCP, problem, and the dual information of the constraint sets 0 < xij < 1 and 0 < yjk I 1 are eXtracted to set the initial values of Lagfrangfian multipliers. Let or, and P, be the dual values associated with the constraint sets 0 < xij < 1 and 0 < yjk I i, TOSpectively. The multipliers t~o and Po of ERSFLCPLR are initialized by the equations t~o = ar, and Po P, We then apply the method described by Fisher (1981) to update the multipliers. At each iteration n, the step size t" is obtained by B", " S= where e1 < B" < 2 e2 6~1, 62 0 ), 8 is the best upper bound of the optimal objective value of ERSFLCPLR found, and 0" is the the optimal objective value of ERSFLCPLR at iteration n. Then, the Lagrangian multipliers are reset by nn"+1pnt = pjk" + s"(nixij .. >;.l, My (1 yjk))Vk The iterations are stopped when the number of iteration exceeds a prespecified value, n,,,, or the upper bound is not improved for umax iterations. 5.5 Computational Study In this section, we present results from our computational study, where we investigate the computational performance of the Lagrangian relaxation approach on the ERSFLCP problem. The computational experiments are implemented on a workstation with two Pentium 4 3.2 GHz processor and 6 GB of memory. 5.5.1 Experimental Design In our study, we conduct experiments on four factors including the maximum number of facilities opened (p), number of capacity levels (K), the incremental amount of beds associated with each capacity level (A), probability of diverting a patient of an opened facility (y), and the monetary value for travel time for a patient (t). For each experiment, we consider three networks consisting of 25, 50 and 100 nodes, respectively, Furthermore, one of the factor is tested at three levels as listed in Table 51, and the other three parameters are set at level 2. Table 51. Experimental factor settings Parameters Level 1 Level 2 Level 3 Experiment 1 p 5 10 20 Experiment 2 K( (A) 5 (8) 8 (5) 10 (4) Experiment 3 y 0.5' 1 2 Experiment 4 t ($/ per minute) 25 50 100 For each experimental setting, we generate 30 random instances. Each random instance is obtained by generating parameters that correspond to patient arrival rates, service rates, fixed and operating costs and distance between each node pair. Specifically, each parameter is obtained by using the formula: u(mean value of the parameter), where u is drawn from the distribution U [0.5, 1.5]. The mean values of the random parameters are listed as follows: * Mean arrival rate (us): 1.5 persons/per hour for each demand node i; * Mean service rate (py): 0.5 persons/per hour for the facility opened at location j; * Mean fixed cost (fj): $5000 /per d~i for the facility opened at location j; * Mean operating cost per bed: $1000 /per d~i for the facility opened at location j; and *Mean distance between each node pair (dij): 50 minutes. Last, the coverage range, d, is set to 50 minutes for all instances. For each instance, the Lagrangian relaxation algorithm described in Section 5.4 is applied. The upper limit of the number of LR iterations (nm=,) is set at 100,000 and the upper limit of the number of consecutive iterations fail to improve the best known feasible solution (um=,) is set at 10,000. The parameter B used to modified the step size is initialized at 2, and divided by 1.5 if the lower bound is not improved for 3NV iterations. 5.5.2 Experimental Results Before presenting the experimental results, we use one of the instances in experiments to show the benefit of utilizing the dual information of ERSFLCP, in LR. The solid lines in Figure 55 depict the upper and lower bounds, respectively, obtained from the LR iterations with the dual information of ERSFLCPr. The dash lines show the LR results without applying the dual information of ERSFLCPr. The dual information of ERSFLCP, provides a good lower bound solution which guides the heuristic to find the best upper bound solution earlier than the guidance of the lower bound obtained from using the Lagfrangfian multipliers generated from scratch. The convergence gaps in Figure 55 are given by x 100, where 8 and 8" are the upper and lower bounds of optimal objective value of ERSFLCPLR obtained from the LR algorithm. Table 52 to Table 55 report the results of four experiments. In general, LR performs quite well in solving the ERSFLCP problem in terms of the convergence gap and the computational effort (i.e., CPU time) required. For the network with 25 nodes, LR takes less 2 seconds to obtain the solutions with convergence gaps less than 5' on average. For the largest network tested, i.e., 100 nodes, the average CPU time is increased, although it is still less than 60 seconds. Moreover, LR does not appear to experience significant increase in convergence gap due to an increase in problem size. I ~I UB (dual) Gap =4% Gap =15% L~4()(()(()( .. LB (dual) ~a" ~~UB (scratch) O 2()(()(()( LB (scratch) () 5()()( 1()(()( 15()()( 2()(()( 25()()( 3()(()( Iteration Figure 55. Convergence speed of the modified LR The first experiment shows that the smaller the value of p is, the higher the CPIT time required to obtain the solution, since smaller p results in a tighter constraint on number of facilities opened. In particular, this trend can he observed in the problems with small and niediumsize networks. Nevertheless, the problem can still be solved within 40 seconds even for largesize networks. Table 52. Experiment 1: effects of nmaxiniun number of facilities opened Network size 25 50 100 Average Gap Time Gap Time Gap Time Gap Time 5 :3.0 1.1 4.4 4.9 5.8 41.8 4.4 15.9 10 :3.1 0.9 :3.0 4.3 :3.8 :35.5 :3.3 1:3.6 20 2.5 0.9 :3.3 :3.2 2.9 29.4 2.9 11.2 Average 2.9 1.0 :3.6; 4.1 4.2 :35.6 :3.5 1:3.6 In the second experiment, we assume that at most 40 heds are used for an opened ER, so the incremental amount of beds per capacity level are changed with the setting of K. A large K value represents that capacity is increased at small scale, i.e., small batch size, and increases the number of variables of the underlying formulation the the ERSFLCP problem. As a result, the CPIT time is increased as K increases, while the growth rate of CPU time is less than the growth rate of K. The impact of K on convergence gap is not significant, especially, with respect to the problem with large size. Table 53. Experiment 2: effects of capacity setting Network size 25 50 100 Average Gap Time Gap Time Gap Time Gap Time K ( (sec.) (sec.) (sec.) (sec.) 5 (8) 0.7 0.8 2.2 4.2 3.6 31.4 2.1 12.1 8 (5) 3.2 1.1 2.9 4.1 3.9 34.0 3.3 13.1 10 (4) 3.6 1.1 3.3 4.5 3.9 35.9 3.6 13.8 Average 2.5 1.0 2.8 4.3 3.8 33.8 3.0 13.0 The third experiment investigates the impact of the diversion probability on the performance of the proposed LR approach. Given the capacity level k, the smaller y incurs the smaller value of the maximum arrival rates that a facility is able to serve, i.e., '., in constraint set (520). That is, the constraint set (520) becomes tighter as y decreases. As shown in Table 54, the CPU time slightly increases as y decreases from 2.01' to 1.'.However, when the value of y is further reduced to 0.5' the required CPU time decreases, which is different from the trend that we observe before. The convergence gap does not appear to be affected by y significantly, in average, the gaps are 3.10' and 2.91' . when y's are 0.5' and 2.01' respectivley. Table 54. Experiment 3: effects of diversion probability Network size 25 50 100 Average Gap Time Gap Time Gap Time Gap Time Y '. (sec.) (sec.) (sec.) (sec.) 0.5 3.7 0.9 3.3 4.1 3.8 34.9 3.6 13.3 1.0 3.1 1.0 3.0 4.3 3.9 56.6 3.3 20.6 2.0 2.4 0.9 2.7 3.6 3.5 48.3 2.9 17.6 Average 3.1 0.9 3.0 4.0 3.7 46.6 3.3 17.2 The last experiment explores the effects of patients' value of time t on the performance of the proposed LR approach. Table 55 shows that the higher the patients' time value is, the smaller the convergence gap is. We also observe that the CPU time is not significantly affected by the change of t. Table 55. Experiment 4: effects of time value Network size 25 50 100 Average Gap Time Gap Time Gap Time Gap Time t ( ) (sec.) (sec.) (sec.) (sec.) 25 4.9 1.1 4.9 5.8 5.6 45.8 5.1 17.6 50 3.0 1.1 2.9 4.7 3.8 49.0 3.2 18.2 100 1.8 1.1 1.7 5.4 2.7 50.8 2.1 19.1 Average 3.2 1.1 3.2 5.3 4.0 48.5 3.5 18.3 To illustrate the performance of the heuristic developed for obtaining the upper bound (feasible) solution at each iteration, we use CPLEX to solve the ERSFLCP problem. The parameters, p, K, y and t are set to their respective values for level 2 in Table 51. For each test instance, CPLEX is stopped if the relative stopping tolerance of 0.01 is satisfied, or CPU time of 3,600 seconds is used. Table 56 summarizes the results, where the relative error of each instance is given by Obj of CPLEX Upper bound of LR x 100I . Obj of CPLEX Table 56 shows that the developed heuristic is very effective. For small and mediansize problems, the upper bounds obtained from the heuristic are very close to the optimal objective values, the average and maximum relative errors are less than 0.;:' and 1.5' . respectively. For the problem with largest size, CPLEX is not able to find the optimal solution within 3,600 seconds. Therefore, we use the best results that CPLEX can find within 3,600 seconds to compare with the results obtained by heuristic. The comparison further ensures the effectiveness of the heuristic developed here. The heuristic not only spends much less time (36 seconds in average) than CPLEX, but also find solutions with better quality in 50I' of the instances. 5.6 Concluding Remarks and Future Research Directions In this chapter, we developed a model for the emergency room services facility location and capacity planning (ERSFLCP) problem, in which each facility is modeled as an M/M/~c/c queueing system to consider the impact of the uncertainties associated with patient arrivals and lengths of stay. The model was designed to simultaneously Table 56. Performance of the heuristic Nodes Instance 1 2 3 4 5 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Min. Avg. Max. 25 CPLEX Heuristic Err. 50 CPLEX Heuristic Err. 100 CPLEX Heuristic Err. (sec.) 0.4 0.7 0.3 0.5 0.8 0.7 0.2 0.2 0.9 0.2 0.8 0.3 0.6 0.3 0.5 0.6 0.8 0.7 0.8 0.4 0.4 0.2 0.3 0.5 0.8 0.6 0.2 0.1 0.1 0.5 0.9 (sec.) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.1 (sec.) 31.8 84.3 44.9 109.9 82.4 47.9 115.0 18.7 27.7 32.6 130.9 65.2 65.5 39.9 25.6 456.4 47.9 34.7 47.3 119.8 53.5 46.7 17.4 62.8 502.1 53.6 27.4 146.6 39.9 176.9 17.4 91.8 502.1 (sec.) 3.3 4.5 3.5 4.7 5.1 7.5 3.5 4.1 3.5 4.6 5.1 5.3 3.0 4.8 3.4 3.7 4.3 3.2 3.5 3.8 3.8 3.4 7.5 3.3 3.5 3.6 3.7 4.8 5.8 4.1 3.0 4.3 7.5 (sec.) >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 >3600 (sec.) 26.6 59.7 26.0 39.7 31.9 33.3 29.4 30.5 51.4 27.8 65.6 30.1 22.8 39.5 25.9 55.3 34.8 41.5 19.5 24.4 25.7 25.4 33.7 49.6 28.1 23.6 19.9 37.1 60.2 46.2 19.5 35.5 65.6 determine the number of facilities opened and their respective locations as well as the capacity levels of the facilities so that the probability of diverting patients is not larger than a particular threshold. A Lagrangian relaxation approach was proposed to obtain solutions of the ERSFLCP problem. The relaxation scheme proposed yield a separable Lagrangian problem that is easy to solve. The upper bound at each LR iteration was obtained by a heuristic developed in this work. In addition, to speed up the convergence process, we II  1h I1 to use the dual information of the linear progranining relaxation of the ER SFLCP problem to generate the initial set of Lagrangian multipliers. The computational study demonstrated that the ERSFLCP problem can he efficiently solved by the Lagrangian relaxation algorithm, and the developed heuristic provides upper bound solutions with good quality. An ininediate extension of our model is to include the closestassignnient constraints which ensure that each demand point is allocated to the closest open facility. Some other practically relevant variations are concerned with the alternative objective functions, such as profit nmaxintization, and travel and service times nxinintization. Another closely related problem is that given some ER service facilities are opened already and B new heds are considered to be added at existing facilities or newly opened facilities. We note that this can he defined by adding the constraints on capacity available, distinguishing the set of existing and new ER locations, and replacing the fixed cost to capacity expansion cost of existing facilities. CHAPTER 6 CONCLUSIONS AND FITTIRE RESEARCH DIRECTIONS In this work, we have presented the integrated use of optimization and queueing theory to determine the optimal capacity plan for health care systems. C'!s Ilter 2 detailed the aggregate hospital bed *r'i' .:; 111.1.rt:e..t (AHBCP) problem and a network flow approach to specify the optimal bed capacity planning decisions throughout a finite planning horizon for hospitals. In this chapter, a hospital was modeled as a G/G/c queueing system with a single bed type and a single patient class. We demonstrated that for realisticsized capacity planning problems, our network formulation is not computationally intensive, and allows us to obtain optimal bed capacity plans quickly. C'!s Ilter 3 introduced the health care team r'/' I 1'1.r t. e.. t. (HC TCP) problem, in which the underlying queueing system was more complex than the one used for AHBCP. In particular, we considered a queueing system where there are two classes of patients and two types of care teams, where service rates are patientclass dependent and one type of care team can substitute for the other. We developed queueing models for both preemptive and nonpreemptive cases, and developed approximation procedures to estimate the average time that each patient class spends in the system. The results from approximation method were then incorporated into the optimization model to determine the minimal cost capacity plan of health care teams throughout a finite planning horizon. Our computational study showed that our approximation approach provides sufficiently accurate results that can he used in practice to make longterm health care team service capacity planning decisions. After the .I_ egate bed capacity was specified in C'!s Ilter 2, we developed the hospital bed allocation (HBA) model to obtain the balanced bed allocation among different medical care units, which were modeled as M/M/~c/c queueing systems. To efficiently solve the problem, we proposed three solution approaches including genetic algorithm(GA), greedy randomized adaptive search procedure (GRASP) and a hybridization of GA and GR ASP (HA). Our computational study showed that the proposed algorithms can solve the problem within a short time while providing solutions with high quality for largesized, realistic test instances. ('!, Ilter 5 developed a emer i. ,...;; room services f r. .:1.:1 location and r'i' I rk'*'":':: T (ERSFLCP) model, where each emergency room service facility is viewed as a Af/Af/c/c queueing system. The model was designed to simultaneously determine the number of facilities opened and their respective locations as well as the capacity levels of the facilities (capture in terms of number of beds) so that the probability that the facility is full and incoming patients have to be diverted (i.e., diversion probability) is not larger than a particular threshold. A Lagrangian relaxation approach was proposed to obtain facility locations and capacity plan. The experimental results illustrated that the Lagrangian relaxation algorithm is very efficient in solving the problem, and the developed heuristic provides solutions with good quality. In our work to date, we primarily focused on strategic level, longterm planning decisions where we made some simplifying assumptions as to the capabilities of the resources and the arrival rates and lengths of stay for the patients. Specifically, we assumed that the available beds in a service (in ('! .pters 3, 4, and 5) or the hospital (in ('!s Ilter 2) are identical. Similarly, we assumed that the arrival rates and length of stay for the patients are identical (in C'!s Ilters 2 and 5). But we also considered the case where patients can he grouped into two classes (each of which corresponds to an acuity level) or into multiple classes (each of which corresponds to a particular speciality) to model arrival rates and lengths of stay. As we mainly focused on strategic level decision 1!! I1:;0s these assumptions are justifiable. However, for more detailed planning there is a need to take other realistic considerations into account. There are typically multiple types of beds (e.g., adult intensive care beds, pediatric intensive care beds, burn intensive care beds, medical/surgical beds), i.e., multiple units, in a service or a hospital. Typically, different units are used to accommodate the patients during different phases of the treatment. Therefore, for more detailed capacity analysis, there is a need to distinguish between these units and model how the patients flow through these units. Another important concern in this context is the consideration of interaction between different units as well as services. That is, if there is no enough capacity in a speciality service, the patient can he accommodated in another speciality service. This, in turn, increases the effective service rate of the speciality and the traffic of another speciality. Similarly, if there is not enough capacity in the downstream unit within a service, then the patient can continue treatment in the upstream unit, i.e., blocking Therefore, for more detailed capacity planning, there is a need to consider multiple types of resources, multiple types of patients, and multiple modes of interaction between units and services. In our work, we have primarily focused on the objective of minimizing total costs (in ('!s Ilters 2, 3, and 5) and balancing work load across units (in ('! .pter 4). Nowced a7 hospitals are focusing on revenue management practices to improve their financial situation. This modern revenue management culture requires hospital administrators to focus on maximizing profit rather than on minimizing operating costs. Therefore, future work should examine the revenue aspects of hospital operations and focus on profit maximization type objectives. However, this shift in emphasis from cost minimization to profit maximization should not ignore the quality aspects associated with the delivery of health care services. In our research, we focused on timeliness (e.g., average service time, average waiting time) and access (e.g., diversion probability) to quantify service quality. Future research in the area should concentrate on developing other metrics to model other aspects of service quality, such as patient safety and service effectiveness. REFERENCES [1] Agency for Healthcare Research and Quality. Last time accessed: November 2007, http://www.ahrq.gov/research/havbed/defintoshm [2] R.K(. Al.0l I T.L. Magnanti, J.B. Orlin, Network Flows: Theory, Algorithms, and Applications, Prentice Hall, New Jersey, 1993. [3] L.A. Aiken, S.P. Clarke, D.M. Sloane et al., Nurses reports on hospital care in five countries, Health Affairs 20 (2001) 4353. [4] L.A. Aiken, S.P. Clarke, D.M. Sloane et al., Hospital nurse staffing and patient mortality, nurse burnout, and job dissatisfaction, JAMA 288 (2002) 19871993. [5] E. Akgahl, M.J. C~tid, C.I Lin, A network flow approach to optimizing hospital bed capacity decisions, Health Care Management Science 9 (2006) 391404. [6] R. Akkerman, M. K~nip, Reallocation of beds to reduce waiting time for cardiac surgery, Health Care Management Science 7 (2004) 119126. [7] American Hospital Association, Hospital Statistics 2005 Edition, Health Forum, Chicago, 2005. [8] P.A. Andersson, E. Varde, F. Diderichsen, Modelling of resource allocation to health care authorities in Stockholm County, Health Care Management Science 3 (2000) 141149. [9] R. Batta, O. Berman, Q. W,v._ Staffing and switching costs in a service center with flexible servers, European Journal of Operational Research 177 (2007) 924938. [10] G.J. Bazzoli, L.R. Brewster, G. Liu, S. K~uo, Does U.S. hospital capacity need to be expanded? Health Affairs 22 (2003) 4054. [11] D. Bellandi, Running at capacity, Modern Healthcare 110 (1999) 112113. [12] P. Beraldi,M.E. Bruni, D. Conforti, Designing robust emergency medical service via stochastic programming, European Journal of Operational Research 158 (2004) 183193. [13] O. Berman, D. K~rass, J. Wang, Locating Service Facilities to Reduce Loss Demand, IIE Transactions 38 (2006) 933946. [14] O. Berman, R. Huang, S. K~im, S., M. Menezes, Locating capacitated facilities to maximize captured demand, IIE Transactions 39 (2007) 10151029. [15] D.M. Berwick, We can cut costs and improve care at the same time, Medical Economics 180 (1996) 185187. [16] G.R. Bitran, D. Tirupati, Capacity planning in manufacturing networks with discrete options, Annals of Operations Research 17 (1989) 119136. [17] G.R. Bitran, D. Tirupati, Tradeoff curves, targeting and balancing in manufacturing queueing networks, Operations Research 37 (1989) 547564. [18] J.T. Blake, M.W. Carter, A goal programming approach to strategic resource allocation in acute care hospitals, European Journal of Operational Research 140 (2002) 541561. [19] J.P.C. Blanc, Performance evaluation of polling systems by means of the powerseries algorithm, Annals of Operations Research 35 (1992) 155186. [20] J.P.C. Blanc, Performance analysis and optimization with the powerseries algorithm, in L. Donatiello, R. Nelson (eds), Performance analysis and optimization with power series algorithm Springer \: 11 I_ Berlin, 5380, 1993. [21] B. Boffey, D. Yates, R.D. Galviio, An algorithm to locate perinatal facilities in the municipality of Rio de Janeiro, Journal of the Operational Research Society 54 (2003) 2131. [22] J. Bowers, B. Lin ix G. Mould, T. Symonds, Modelling outpatient capacity for a diagnosis and treatment centre, Health Care Management Science 8 (2005) 205211. [23] J. Bowers, G. Mould, Managing uncertainty in orthopaedic trauma theatres, European Journal of Operational Research 154 (2004) 599608. [24] J. Bowers, G. Mould, Ambulatory care and orthopaedic capacity pI l ....ilr_ Health Care Management Science 8 (2005) 4147. [25] M.L. Brandeau, F. Sainfort, W.P. Pierskalla, Operations Research and Health Care: A Handbook of Methods and Applications, Springer, Boston, 2004. [26] K(.M. Bretthauer, M.J. Ci~ti, A model for planning resource requirements in health care organizations, Decision Sciences 29 (1998) 243270. [27] M.J. Brusco, M.J. Showalter, Constrained nurse staff 2.1, l1i; OMEGA: International Journal of Management Science 21 (1993) 175186. [28] E.K(. Burke, P. De Causmaecker, G. Vanden Berghe, H. Van Landeghem, The state of the art of nurse 10 ~ha li, Journal of Scheduling 7 (2004) 441499. [29] F. Cerne, J. Montague, Capacity crisis, Hospitals and Health Networks 68 (19) (1994) 3036. [30] J.K(. Cochran, A. Bharti, Stochastic bed balancing of an obstetrics hospital, Health Care Management Science 9 (2006) 3145. [31] R.C. Coile Jr., Futurescan 2002: a forecast of healthcare trends 20022006, Health Administration, Chicago, 2002. [32] M.J. Ci~ti, S.L. Tucker, Four methodologies to improve healthcare demand forecasting, Healthcare Financial Management 55 (2001) 5458. [:33] R. Davies, Simulation for planning services for patients with coronary artery disease, European Journal of Operational Research 72 (1994) :32:3332. [:34] K(. Davis, S.C. Schoenhaunt, A.M. Audet, A 2020 Vision of PatientCentered Primary Care, Journal of General Internal Medicine 20 (2005) 95:3957. [:35] L. Delesie, A. K~astelein, F. Merode, J.M.H. Vissers, Managing health care under resource constraints, European Journal of Operational Research 105 (1998) 247247. [:36] T.A. Feo, M.G.C. Resende, A probabilistic heuristic for a computationally difficult set covering problem, Operations Research Letters 8 (1989) 6771. [:37] M.L. Fisher, The Lagrangian relaxation method for solving integer progranining problems, Alanagenient Science 27 (1981) 118. [:38] S. Flessa, Where efficiency saves lives: A linear progranine for the optimal allocation of health care resources in developing countries, Health Care Alanagenient Science :3 (2000) 249267. [:39] R.D. Galvao, L.G.A. Espejo, B. Boffey, A hierarchical model for the location of perinatal facilities in the municipality of Rio de Janeiro, European Journal of Operational Research 1:38 (2002) 495517. [40] R.D. Galviio, L.G.A. Espejo, B. Boffey, D. Yates, Load balancing and capacity constraints in a hierarchical location model, European Journal of Operational Research 172 (2006) 6:31646. [41] D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, AddisonWesley, Wokingham, England, 1989. [42] S.M. Goldstein, P.T. Ward, G.K(. Leong, T.W. Butler, The effect of location, strategy, and operations technology on hospital performance, Journal of Operations Alanagenient 20 (2002) 6:375. [4:3] F. Gorunescu, S.I. McClean, P.H. Millard, Using a queueing model to help plan bed allocation in a department of geriatric medicine, Health Care Alanagenient Science 5 (2002) :307312. [44] L. Green, A queuing system with generaluse and linliteduse servers, Operations Research :33 (1985) 168182. [45] L.V. Green, How many hospital beds? Inquiry :38 (2002/200:3) 400412. [46] L.V. Green, V. Nguyen, Strategies for cutting hospital beds: the impact on patient service, Health Services Research :36 (2001) 421442. [47] S. Groothuis, A. Hasnian, P.E.J. van Pol et al., Predicting capacities required in cardiology units for heart failure patients via simulation, Computer methods and Programs in Biontedicine 74 (2) (2004) 129141. [48] D. Gross, C.M. Harris, Fundamentals of Queueing Theory (3nd ed.), John Wiley and Sons, New York, 1998. [49] K(. Grumbach, T. Bodenheimer, Can health care teams improve primary care practice? JAMA: the journal of the American Medical Association 291 (2004) 12461251. [50] C. Hayward, What are we going to do with all our excess capacity? Health Care Strategic Management 16 (1998) 2023. [51] P.R. Harper, A framework for operational modelling of hospital resources, Health Care Management Science 5 (2002) 165173. [52] G.W. Harrison, A. Shafer, M. Mackay, Modelling Variability in Hospital Bed Occupancy, Health Care Management Science 8 (2005) 325334. [53] J.H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, 1975. [54] G. Hooghiemstra, M. K~eane, S.V.D. Ree, Power series for stationary distributions of coupled processor models, SIAM Journal on Applied Mathematics 48 (1988) 11591166. [55] Institute of Medicine, Crossing the Quality ChI Ims! National A< II1. ...vJ Press, Washington, D.C, 2001. [56] H. Jia, F. Ordonez, M.M. Dessouky, A modeling framework for facility location of medical services for largescale emergencies, IIE Transactions 39 (2007) 4155. [57] A.M. Johnson, Capacity planning for the future, Journal of Health Care Finance 24 (1997) 7275. [58] J.B. Jun, S.H. Jacobson, J.R. Swisher, Application of discreteevent simulation in health care clinics: A survey, Journal of the Operational Research Society 50 (2) (1999) 109123. [59] E.P.C. K~ao, C. Lin, The M/M/1 queue with randomly varying arrival and service rates: a phase substitution solution, Management Science 35 (1989) 561570. [60] E.P.C. K~ao, K(.S. Narayanan, Computing steadystate probabilities of a nonpreemptive priority multiserver queue, ORSA Journal on Computing 2 (1990) 211218. [61] E.P.C. K~ao, M. Queyranne, Budgeting costs of nursing in a hospital, Management Science 31 (1985) 608621. [62] E.P.C. K~ao, G.G. Tung, Aggregate nursing requirement planning in a public health care delivery system, SocioEconomics Planning Science 15 (1981) 119127. [63] E.P.C. K~ao, G.G. Tung, Bed allocation in a public health care delivery system, Management Science 27 (1981) 507520. [64] E.P.C. K~ao, S.D. Wilson, Analysis of nonpreemptive priority queues with multiple servers and two priority classes, European Journal of Operational Research 118 (1999) 181193. [65] E.H. K~aplan, M. Johri, Treatment on demand: an operational model, Health Care Management Science 3 (2000) 171183. [66] S. K~avanagh, J. Cowan, Reducing risk in healthcare teams: an overview, Clinical Governance 9 (2004) 200204. [67] S. K~im, I. Horowitz, K(.K. Young, T.A. Buckley, Analysis of capacity management of the intensive care unit in a hospital, European Journal of Operational Research 115 (1999) 3646. [68] S. K~im, I. Horowitz, K(.K. Young, T.A. Buckley, Flexible bed allocation and performance in the intensive care unit, Journal of Operations Management 18 (2000) 427443. [69] L. K~leinrock, Queueing systems: Theory Vol. 1, Wiley, New York, 1975. [70] N. K~oizumi, E. K~uno, T.E. Smith, Modeling patient flows using a queuing network with blocking, Health Care Management Science 8 (2005) 4960. [71] G. K~oole, On the use of the power series algorithm for general Markov processes, with an application to a Petri net, INFORMS Journal on Computing 9 (1997) 5156. [72] D.H. K~ropp, R.C. Carlson, Recursive modeling of ambulatory health care settings, Journal of Medical Systems 1 (1977) 123135. [73] R.J. K~usters, P.M.A. Groot, Modelling resource availability in general hospitals design and implementation of a decision support model, European Journal of Operational Research 88 (1996) 428445. [74] G. Latouche, V. Ramaswami, A logarithmic reduction algorithm for quasibirthanddeath processes, Journal of Applied Probability 30 (1993) 650674. [75] G. Latouche, V. Ramaswami, Introduction to Matrix Analytic Methods, SIAM, Philadelphia, PA, 1999. [76] L.X. Li, W.C. Benton, Performance measurement criteria in health care organizations: Review and future research directions, European Journal of Operational Research 93 (1996) 449468. [77] L.X. Li, W.C. Benton, G.K(. L ..nr The impact of strategic operations management decisions on community hospital performance, Journal of Operations Management 20 (2002) 389408. [78] E. Litvak, P.I. Buerhaus, F. Davidoff, F. et al., Managing unnecessary variability in patient demand to reduce nursing stress and improve patient safety, Joint Coninission Journal on Quality and Patient Safety :31 (2005) :330:338. [79] 31. Mackay, Practical experience with bed occupancy nianagenient and planning systems: an Australian view, Health Care Alanagenient Science 4 (2001) 4756. [80] 31. Mackay, 31. Lee, ClI u! e. of Models for the Analysis and Forecasting of Hospital Beds, Health Care Alanagenient Science 8 (2005) 221230. [81] V. Alarianov, C. ReVelle, The queueing nmaxinmal availability location problem: A model for the siting of emergency vehicles, European Journal of Operational Research 9:3 (1996) 110120. [82] A.H. Marshall, S.I. McClean, C.M. Shapcott, P.H. Millard, Modelling patient duration of stay to facilitate resource nianagenient of geriatric hospitals, Health Care Alanagenient Science 5 (2002) :31:3319. [8:3] A.H. Marshall, C. Vasilakis, E. ElDarzi, Length of r lihased patient flow models: recent developments and future directions, Health Care Alanagenient Science 8 (2005) 21:3220. [84] B.J. Masterson, T.G. Mihara, G. Miller, S.C. Randolph, M.E. Forkner, A.L. Crouter, Using models and data to support optimization of the military health system: A case study in an intensive care unit, Health Care Alanagenient Science 7 (2004) 217224. [85] W.E. McAleer, I.A. Naqvi, The relocation of ambulance stations: A successful case study, European Journal of Operational Research 75 (1994) 582588. [86] L.W. Morton, N. Bills, D. K~ay, Boosting local economies through health care economic development, Last time accessed: August 2005, http://www.cardi. cornell.edu/ economic c_development/community_economic_renewal/007.php Coninunity and Rural Development Institute, Cornell University. [87] M.F. Neuts, MatrixGeonietric Solutions in Stochastic Models, Johns Hopkins University Press, Maryland, 1981. [88] J.M. Nguyen, P. Six, T. Ch1 II1 Il.t, D. Antonioli, P. Lonthrail, P. Le Beux, An objective method for bed capacity planning in a hospital department A comparison with target ratio methods, Methods of information in medicine 46 (4) (2007) :399405. [89] W.P. Pierskalla, Health care delivery, Presented at the National Science Foundation Workshop on Engineering the Service Sector, Atlanta, 2001. [90] W.P. Pierskalla, D. Brailer, Applications of operations research in health care delivery, in Beyond the profit motive: public sector applications and methodology, Handbooks in OR&MS, S. Pollock, A. Barnett, M. Rothkopf (eds), Vol 6, NorthHolland, New York, 1994. [91] W.P. Pierskalla, D. Wilson, Review of operations research improvements in patient care delivery systems, Working paper, University of Pennsylvania, Philadelphia, 1989. [92] S. Rahman, D.K(. Smith, Use of locationallocation models in health service development planning in developing nations, European Journal of Operational Research 123 (2000) 437452. [93] J.F. Repede, J.J. Bernardo, Developing and validating a decision support system for locating emergency medical vehicles in Louisville, K~entucky, European Journal of Operational Research 75 (1994) 567581. [94] J.C. Ridge, S.K(. Jones, M.S. Nielsen, A.K(. Shahani, Capacity planning for intensive care units, European Journal of Operational Research 105 (1998) 346355. [95] S.M. Ryan, Capacity expansion for random exponential demand growth with lead times, Management Science 50 (2004) 740748. [96] F. Sainfort, Where is OR/ ilS in the present crises in health care delivery? Presented at the Institute for Operations Research and the Management Sciences Annual Meeting, Miami, 2001. [97] D.P. Schneider, K(.E. K~ilpatrick, An Optimum Manpower Utilization Model for Health Maintenance Organizations, Operations Research 23 (1975) 869889. [98] L. Shi, D.A. Singh, Essentials of the US Health Care System, Jones and Bartlett Publishers, Maryland, 2005. [99] R.A. Shumsky, Approximation and analysis of a queueing system with flexible and specialized servers, OR Spectrum, Special Issue on Call Center Management 26 (3) (2004) 307330. [100] D. Sinreich, Y. Marmor, Emergency department operations: the basis for developing a simulation tool, IIE Transactions 37 (3) (2005) 233245. [101] V.L. SmithDaniels, S.B. Schweikhart, D.E. SmithDaniels, Capacity management in health care services: review and future research directions, Decision Science 19 (1988) 899918 [102] C. Smith, C. Cowan, A. So1., 1.1_ A. Catlin, Health Spending Growth Slows in 2004, Health Affairs 25 (2006) 186196. [103] L. Snyder, Facility location under uncertainty: a review, IIE Transactions 38 (2006) 537554. [104] L. Snyder, 31. Daskin, Reliability Models for Facility Location: The Expected Failure Cost Case, Transportation Science 39 (3) (2005) 400416. [105] D.A. Stanford, W.K(. Grassniann, The bilingual server system: a queueing model featuring fully and partially qualified servers, INFOR 31 (4) (1993) 261277. [106] J. Thorson, Last time accessed: August 2006, http://sepwww.stanford.edu/oldreports/sep2/01_b~t1 [107] United States Department of Labor Bureau of Labor Statistics, Last time accessed: August 2006, http://www.bls. gov/oco/cg/cgs035 .htm#emply. [108] 31. IUtley, S. Gallivan, K(. Davis, P. Daniel, P. Reeves, J. Worrall, Estimating bed requirements for an intermediate care facility, European Journal of Operational Research 150 (2003) 92100. [109] A.H. van Zon, G.J. K~oniner, Patient flows and optimal healthcare resource allocation at the macrolevel: a dynamic linear progranining approach, Health Care Alanagenient Science 2 (1999) 8796. [110] J.M.H. Vissers, Patient flowbased allocation of inpatient resources: A case study, European Journal of Operational Research 105 (1998) 356370. [111] T.D. Vries, R.E. Beeknlan, Applying simple dynamic modelling for decision support in planning regional health care, European Journal of Operational Research 105 (1998) 277284. [112] S. Yan, Approximating Reduced Costs under Degeneracy in a Network Flow Problem with Side Constraints, Networks 27 (1996) 267278. BIOGRAPHICAL SKETCH ChinI Lin was born in Taipei, Taiwan. She received her B.S. and 1\.S in civil engineering from the National Central University in Taiwan in 1994 and 1996, respectively. From 1997 to 2002, she worked for Cl....! I Airlines, and her 1!! li r~ tasks included demand forecast, market analysis, route analysis, and fleet planning. She pursued her master and doctoral degrees in the Department of Industrial and Systems Engineering at the University of Florida since 2002. ChinI's main research interest is Operations Research, and topics of special interest are health care management and airline flight/crew scheduling. Thus far, her research has focused on capacity management in health care delivery. PAGE 1 1 PAGE 2 2 PAGE 3 3 PAGE 4 Iwouldliketothankallpeoplewhohavehelpedandinspiredmeduringmydoctoralstudy.Iwanttoexpressmysinceregratitudetomydissertationadvisor,Dr.ElifAkcal,forherguidance,insightandsupportduringthisresearchandstudy.Iamalsogratefultomycommitteemembers,Dr.FaridAitSahlia,Dr.P.OscarBoykinandDr.SiriphongLawphongpanich,fortheirconstructivesuggestionsandcomments.Iwishtoextendmywarmestthankstomymentor,Dr.ShangyaoYan,forleadingmetotheeldofoperationsresearch,hisfriendshipandnumerousfruitfuldiscussions.Mydeepestgratitudegoestotomyparents,ZheXiongandFangXue,andmyhusband,ChungJui.Withouttheirunderstandingandencouragement,itwouldhavebeenimpossibleformetocompletemydegree. 4 PAGE 5 page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 10 2AGGREGATEHOSPITALBEDCAPACITYPLANNING ............ 13 2.1Introduction ................................... 13 2.2LiteratureReview ................................ 15 2.3ProblemFormulation .............................. 17 2.3.1RestrictedBedCapacityPlanningProblem .............. 19 2.3.2RestrictedBedCapacityPlanningProblemwithShuttering ..... 22 2.4IllustrationoftheModel ............................ 25 2.4.1ARepresentativeDecisionMakingScenario .............. 25 2.4.2Experiment1{AnApplicationoftheModel ............. 26 2.4.3Experiment2{AssessingtheImpactofProblemParameters .... 29 2.5Extensions .................................... 33 2.6ConcludingRemarksandFutureResearchDirections ............ 34 3HEALTHCARETEAMCAPACITYPLANNING ................ 37 3.1Introduction ................................... 37 3.2LiteratureReview ................................ 42 3.3ProblemFormulation .............................. 44 3.4QueueingAnalysis ............................... 47 3.4.1PreemptiveCase:EmergencyMedicineServices ........... 48 3.4.2NonPreemptiveCase:OutpatientClinicServices .......... 51 3.5ComputationalStudy .............................. 56 3.5.1ComputationalPerformanceoftheDBAMethod ........... 57 3.5.2ComputationalPerformanceoftheHCTSCPModel ......... 62 3.6ConcludingRemarksandFutureResearchDirections ............ 66 4HOSPITALBEDALLOCATIONPROBLEM ................... 69 4.1Introduction ................................... 69 4.2LiteratureReview ................................ 70 4.3ProblemFormulation .............................. 73 4.4SolutionAlgorithms .............................. 75 5 PAGE 6 ............................ 75 4.4.2GreedyRandomizedAdaptiveSearchProcedure ........... 80 4.4.3HybridizationofGA&GRASP .................... 81 4.5ComputationalStudy .............................. 83 4.5.1SummaryofResultsObtainedbyGA ................. 85 4.5.2SummaryofResultsObtainedbyGRASP .............. 88 4.5.3SummaryofResultsObtainedbyHA ................. 88 4.6ConcludingRemarksandFutureResearchDirections ............ 92 5EMERGENCYROOMSERVICESFACILITYLOCATIONANDCAPACITYPLANNING ...................................... 94 5.1Introduction ................................... 94 5.2LiteratureReview ................................ 95 5.3ProblemFormulation .............................. 97 5.4SolutionApproach ............................... 100 5.4.1LagrangianRelaxationApproach ................... 100 5.4.2LowerBound .............................. 101 5.4.3UpperBound .............................. 103 5.4.4LagrangianMultipliers ......................... 106 5.5ComputationalStudy .............................. 107 5.5.1ExperimentalDesign .......................... 107 5.5.2ExperimentalResults .......................... 108 5.6ConcludingRemarksandFutureResearchDirections ............ 111 6CONCLUSIONSANDFUTURERESEARCHDIRECTIONS .......... 114 REFERENCES ....................................... 117 BIOGRAPHICALSKETCH ................................ 125 6 PAGE 7 Table page 21Parametersettingsforthebasescenario,S1 .................... 27 22Scenariodescriptionsforexperiment1 ....................... 27 23SummarystatisticsfortheRBCPwSproblem'ssolutiontime(inCPUseconds)asafunctionofinitialeectivecapacity ...................... 31 24SummarystatisticsfortheRBCPwSproblem'ssolutionasafunctionofcapacitylevelsandthelengthoftheplanninghorizon .................... 32 31Thepossibletransitionsenterstate(n1;n2;m)fortheEDsetting ........ 49 32ComputationalrequirementoftheDBA,AGE,andGEmethods ......... 58 33Relativeandpercentageerror:preemptivecase .................. 60 34Relativeandpercentageerror:nonpreemptivecase ................ 61 35Teamcongurations ................................. 63 36Parametersettings .................................. 63 37Impactofthefractionofclass1patientsonthesimilarityindex ......... 65 38Impactofunitpatientdelaycostonthesimilarityindex ............. 65 39Impactofmaximumallowableaveragetimeinsystemonthesimilarityindex .. 65 41NearoptimalsolutionsobtainedbyusingCPLEX ................. 86 42SolutionsobtainedbyGA .............................. 87 43SolutionsobtainedbyGRASP ............................ 89 44SolutionsobtainedbyHA .............................. 91 51Experimentalfactorsettings ............................. 107 52Experiment1:eectsofmaximumnumberoffacilitiesopened .......... 109 53Experiment2:eectsofcapacitysetting ...................... 110 54Experiment3:eectsofdiversionprobability ................... 110 55Experiment4:eectsoftimevalue ......................... 111 56Performanceoftheheuristic ............................. 112 7 PAGE 8 Figure page 21NetworkowrepresentationforRBCPwithc0=300,B=25,n=1,andT=4 ... 21 22NetworkowrepresentationforRBCPwSwithc0=275,B=25,n=1,andT=4 24 23Patientarrivalrateforexperiment1 ........................ 28 24Optimalcapacityplansforexperiment1 ...................... 28 25Numberofnodesinthenetworkasafunctionofinitialeectivebedcapacity .. 30 31AnillustrationofthenetworkrepresentationforHCTSCP ............ 47 32TwodimensionalCTMCapproximation ...................... 52 41Pseudocodeofthegeneticalgorithm ........................ 76 42Pseudocodeofthepopulationgeneratingprocedure ................ 77 43Pseudocodeofoccupancydrivenallocation .................... 77 44Pseudocodeofrandomrectiedprocedure ..................... 78 45Pseudocodeofcrossoverprocedure ......................... 79 46Exampleofcrossover ................................. 79 47Pseudocodeofthemutationprocedure ....................... 80 48PseudocodeofGRASP ............................... 80 49Pseudocodeofgreedyrandomizedconstructionprocedure ............ 82 410Pseudocodeoflocalsearchprocedure ........................ 83 411PseudocodeofHA .................................. 83 412Pseudocodeofelitesetgenerationprocedure ................... 84 413Theupdatingfunctionsof 90 51PseudocodeoftheLagrangianrelaxation ..................... 101 52Pseudocodeofthefeasiblesolutiongeneration ................... 104 53Pseudocodeofcoveringalldemandnodes ..................... 105 54Pseudocodeofclosingfacilities ........................... 105 55ConvergencespeedofthemodiedLR ....................... 109 8 PAGE 9 Healthcarecapacityplanningistheartandscienceofpredictingthequantityofresourcesrequiredtodeliverhealthcareserviceatspeciedlevelsofcostandquality.Becauseofvariabilityinthearrivalofpatientsandinthedeliveryofhealthcareservices,successfullymeetingthedemandforhealthcareservicesisadauntingtaskthatrequiresanunderstandingoftheinherenttradeobetweenitscostandqualityofservice. Inourwork,wemodelthegeneralhealthcaresystemsasqueueingstationsandincorporatequeueingtheoryintoanoptimizationframework.Thequeueingmodelingapproachcapturesthestochasticnatureofarrivalsandservicetimesthatistypicalinhealthcaresystems.Theoptimizationframeworkdeterminestheminimumcostcapacityrequiredtoachieveatargetlevelofcustomerservice.Theinclusionsofqueueingequationsanddiscretecapacityoptionsresultthecapacityplanningmodelsinnonlinearintegerprogrammingformulations. Wedevelopeectivesolutionalgorithmstoobtainhighqualitysolutionsparticularlyforrealisticsizedproblems.Fortheanalysisofunderlyingqueuingsystems,weeitheruseavailableresultsfromtheliteratureordevelopapproximations.Forthesolutionofoptimizationmodels,weemploynetworkoptimization,metaheuristic,andLagrangianrelaxationapproachestodevelopeectivesolutionalgorithms.Wepresentresultsfromextensivecomputationalexperimentstodemonstratethecomputationaleciencyandeectivenessoftheproposedsolutionapproaches. 9 PAGE 10 Capacityplanningdecisionsareimportanttoanyindustry,especiallytohealthcareindustrybecausenotonlyitrelatestothemanagementofhighlyspecializedandcostlyresources(i.e.,nurses,doctors,andadvancedmedicalequipment),butalsoitmakesadierencebetweenlifeanddeathincriticalconditions.Healthcarecapacityplanninginvolvespredictingthequantityandparticularattributesofresourcesrequiredtodeliverhealthcareserviceatspeciedlevelsofcostandquality. Accordingtotheresourceavailabilities,capacityplanningcanbeclassiedintothreelevelsincludingstrategic,tacticalandoperationallevels.Inthestrategiclevel,decisionmakersfocusonthelongtermcapacitydecisionssuchaslocationsofmedicalfacilitiesorsizesofmedicalfacilitiesandworkforce.Thetacticallevelcapacityplanconcernspolicieswhichcouldimprovetheserviceperformance,ontheperspectiveofeitherhealthcareprovidersorreceivers,byreallocating,expandingordownsizingthecurrentresources.Incontrast,thecapacityplanontheoperationallevelconcentratesonhowtomeetdemandbyusingtheexistingresourcesthroughappropriatemethodssuchasschedulingoroverrun. Inourstudy,wefocusonlongtermlevelhealthcarecapacityplanningdecisions,wheretheknowledgeoftheperformanceofthesystematsteadystateissucient.Inaddition,wemodelthehealthcareservicefacilitiesasqueueingsystemstotakethestochasticpatientarrivalsandlengthofstayintoaccountandmeasurethesystemperformance.Thepurposeofourstudyistoincorporatequeueingmodelsintooptimizationframeworktodeterminetheoptimalcapacitylevelthatminimizescostwhilemaintainingadesiredlevelofperformanceonpatients'servicequalityandnancialand/oroperationalrestrictions. Inourwork,westudyfourdierent,practicallyrelevant,longtermcapacityplanningproblems.Chapter 2 introducestheaggregatehospitalbedcapacityplanning(AHBCP) 10 PAGE 11 16 17 ]toestimatethepatients'expectedwaitingtime,wedeterminetheoptimalbedcapacityplanoveraniteplanninghorizonutilizinganetworkowapproach. Chapter 3 presentsthehealthcareteamcapacityplanning(HCTCP)problem,whichndsapplicationsinhospitalemergencyroomandoutpatientclinicalsettings.TheunderlyingqueueingsystemismorecomplexthantheoneusedforAHBCP.Inparticular,weconsiderasystemwithtwoclassesofpatientswithdierentprioritiesandtwotypesofhealthcareteamswithpatientclassdependentservicerates.Moreover,thereisasymmetricsubstitutabilitybetweentheteams,i.e.,whileoneteamcanprovideservicetobothclassesofpatient,whereastheotherteamcanserveonlyoneclassofpatient.Forthisqueueingsystem,wedevelopanapproximationapproachtocomputetheaveragetimethatapatientspendsinthesystemforeachpatientclass.Then,weintegratetheresultsfromapproximationmethodintoanoptimizationmodeltomakelongtermhealthcareteamcapacitydecisions. Chapter 4 statesthehospitalbedallocation(HBA)problem,whichisanextensiontheAHBCPproblem.Aftertheaggregatebedcapacityisspecied,thenextstepinvolvedisconcernedwiththeallocationofaggregatebedcapacityamongdierentmedicalcareunits(MCUs).WemodeleachMCUinahospitalasaM=M=c=cqueueingsystemtoestimatetheprobabilityofrejectionwhentherearecbedsintheunit.WedevelopanoptimizationmodeltoallocatetheaggregatebedcapacityacrossdierentMCUs. Chapter 5 detailstheemergencyroomservicesfacilitylocationandcapacityplanning(ERSFLCP)problem,inwhicheachfacilityismodeledasaM=M=c=cqueueingsystem.Weconstructafacilitylocationmodel,whichsimultaneouslydeterminesthenumberoffacilitiesopenedandtheirrespectivelocationsaswellasthecapacitylevelsofthefacilitiessothattheprobabilitythatallserversinafacilityarebusydoesnotexceedapredeterminedlevel. 11 PAGE 12 6 providesasummaryofthefourproblemswehaveinvestigatedandsuggestsfutureresearchdirections. 12 PAGE 13 Whilecapacityplanninghaschallengedhealthcaredecisionmakersandresearchersfordecades[ 90 91 101 ],thereisarenewedsenseofurgencytoaddressthisproblem.Inadditiontotheperennialstrugglebetweenthecontinuallyincreasingcostsofhighlyspecializedandscarceinputs(i.e.,skilledandexiblesta,advancedclinicalandmedicaltechnologyandequipment,physicalspaceandsupplies)anddeclininggovernmentandprivatereimbursements[ 89 96 ],thedemandforinpatientcarehasbeengrowingsubstantially.AccordingtotheAmericanHospitalAssociation(AHA),whileaveragelengthofstay(ALOS)remainedunchangedat5.7days,allcommunityhospitalvolumestatisticsincreasedfrom2002to2003:inpatientadmissionsby0.9%to34.8million, 13 PAGE 14 7 ].However,thenumberofhospitalsofalltypesdecreasedby30to5,764,therewere32fewercommunityhospitals,and8,000fewercommunityhospitalbedsin2003[ 7 ]. Inthispaper,wefocusonaggregatehospitalbedcapacityplanningdecisions.Wedevelopamodeltosimultaneouslydeterminethetimingandmagnitudeofchangesinbedcapacitythatminimizescapacitycost(includingthecostofchangingcapacityaswellasthecostofoperatingcapacity)whilemaintainingadesiredleveloffacilityperformance(e.g.,limitingapatient'sexpecteddelaybeforebeingadmittedtoabedandkeepingexpenseswithinbudget)overaniteplanninghorizon.Wedividetheplanninghorizonintodiscretetimeperiodsofequallength,andassumethatthesystemachievessteadystateineachoftheseintervals.Thisallowsustousequeuingmethodologytoanalyzesystemperformance,butthistypicallyleadstononlinearequationsinourformulation.Ashospitalbedcapacitymustbeintegervalued,ourplanningmodelisalargescalenonlinearintegeroptimizationmodelthatminimizestotalcostwhileachievingatargetedlevelofsystemperformance.Weshowthatsomepracticalconsiderationsleadtosimplicationsinthemodel,whichleadstoanetworkowformulationfortheproblemthatcanbesolvedinpolynomialtime. Avarietyofproblemsthatariseinthecontextoftransportation,nance,manufacturing,andservicesystemscanbemodeledasnetworkowmodels[ 2 ].Anetworkisacollectionof(capacitatedoruncapacitated)nodesand(directed/undirectedandcapacitated/uncapacitated)arcs,wherethearcslinkonenodetoanotherandcarryowfromonenodetoanother.Wellknownnetworkowmodelsaretheshortestpath,maximumow,andminimumtotalcostowformulations,forwhichecientsolutionalgorithmsexist[ 2 ].Inourwork,weshowthatthecapacityplanningmodelweconsidercanbetransformedintoashortest 14 PAGE 15 Theremainderofthispaperisorganizedasfollows.Section 2.2 providesabriefoverviewofthehistoryandcurrentresearchinhospitalbedplanning.InSection 2.3 ,wedescribethesystemandgivethreemodelsforplanninghospitalbedcapacity.InSection 2.4 ,usingdatafromamediumsizedmedicalcenter,weprovideacomputationalstudytoillustratehowthemodelformulationscanbeusedandhowchangesinproblemparameterscanaectourabilitytoobtainanoptimalsolution.Section 2.5 ,oersseveralpracticalextensionsofourmodel.Last,wegiveconcludingremarksanddiscussfutureresearchdirectionsinSection 2.6 10 15 29 45 46 50 ].Lessthanadecadelater,inpartduetorenewedgrowthindemandforinpatientservices[ 7 31 ],mosthospitalsarecurrentlyfacingconsiderablespaceandresourcerestrictionsforcingthemtocontemplateexpensiverenovationsand/ornewconstructionprojectstoincreasebedcapacity[ 11 31 57 ].However,whetherhospitals,infact,needtheadditionalcapacityappearstobeunresolved[ 10 45 ].Ononehand,increasedinpatientadmissionscoupledwithfewerhospitalsandfewerhospitalbedswouldsupporttheargumentinfavorofcapacityincreases[ 7 31 ].Conversely,levelordecreasingaveragelengthofstayandacorrespondingdecreaseintheaverageinpatientoccupancyratemayimplythatexistingcapacityissucient[ 10 ].Regardless,determiningtheoptimalnumberandorganizationofhospitalbedscontinuestobeachallenge. Theabilitytoanticipatebeddemandandmatchitwiththeappropriatebedsupplyiscriticaltoeectivebedplanning.Healthcaredecisionmakersknowthatbothwillbeinuencedbyanumberoffactors.Factorsinternaltothedecisionmakersinclude 15 PAGE 16 45 46 ].Externally,factorsfacingdecisionmakersincludeatypicalchangesincommunityhealth(e.g.,severeustrains),annualholidays(e.g.,Thanksgiving),andtheavailability,size,andcompositionofappropriatemedicalpersonnel.Historically,startingwiththeHill{BurtonActof1946,bedcapacityplanninghastendedtobebasedontargetoccupancylevels(TOLs)thatareassumedtoreectcapacitylevelsthatachieveanappropriatebalanceofcost,patientdelays,andresourceutilization.TOLsarederivedusinganalyticmodelsoftypicalhospitalsindierentcategoriesandarebasedonacceptablepatientdelaysfordierentservices.However,GreenandNguyen[ 46 ]usequeuingmodelstoinvestigatetherelationshipbetweenoccupancylevelsanddelay,andconcludedthatusingTOLsastheprimarydeterminantofbedcapacityisinadequateandmayleadtoexcessivedelaysforbeds.Inparticular,aTOLdoesnotnecessarilycorrespondtoadesiredservicelevel,andthereisaneedtoquantifythedesiredservicelevelandmeasureitscostimplicationsaccurately. Ryan[ 95 ]providesacapacityexpansionmodelwithexponentialdemandandcontinuoustimeintervalsandcontinuousfacilitysizes.Inthecontextofhealthcareplanning,however,itismorerealistictomodelcapacityexpansionastheproductoflimited,discretechoicesasroutineplanningsessions(e.g.,bimonthlyorquarterly)wherecapacityincreasesordecreasesoccurinsomexedbedamountsuchasa20bedunit.BretthauerandC^ote[ 26 ]modelageneralhealthcaredeliverysystemasanetworkofqueuingstationsandincorporatethequeuingnetworkintoanoptimizationframeworktodeterminetheoptimalcapacitylevelssubjecttoaspeciedlevelofsystemperformance 16 PAGE 17 7 ],theactualtforagivenfacilitywillnotbeknownuntilthedemandpresentsitself.Therefore,forthepurposesofcapacityplanning,tcanbeforecastedbyaseasonallyadjustedtrendline,forexample[ 32 ].Lettdenotethemaximumallowableexpecteddelayforapatientbeforethepatientisadmittedtoabedinperiodt.Wenotethatthenumberofbedsinthesysteminagivenperiodcanbelimitedduetootherresourcelimitationsincludingasthephysicalsizeofthefacilityand/ortheamountandtypeofpersonnelavailable.Letc0betheinitialbedcapacityinthehospital.Last,thereisabudgetlimitontheamountofmonetaryresourcesthatcanbeallocatedtopurchasingadditionalbedcapacitydenotedbyt. Wehavethreetypesofdecisionvariables.Letxtbenumberofbedsinperiodt.Letx+tbetheamountofincreaseinbedcapacityatthebeginningofperiodt,andxtthe 17 PAGE 18 minTXt=1f(xt;t;t)+TXt=1g(xt1;xt)+TXt=1h(xt) (2{1)subjecttow(xt;t;t)t8t Theobjectivefunction( 2{1 )minimizesthetotalcostofpatientwaiting,changingthebedcapacity,andoperatingtheexistingbedcapacity.Constraint( 2{2 )imposesamaximumallowablelimitontheexpectedpatientwaiting.Forexample,inordertoquantifytheexpecteddelayforapatienttobeadmittedtoabed,weassumethatthehospitalcanberepresentedasaGI=G=squeueingsystemandusetheexpectedwaitingtimeapproximationprovidedbyBitranandTirupati[ 16 17 ]tocalculateapatient'sexpectedwaitforahospitalbed.Constraint( 2{3 )setstheinitialbedcapacitywhileconstraint( 2{4 )isaowbalanceequationstatingthatthenumberofbedsavailableinaperiodisequaltothenumberofbedsavailableinthepreviousperiodplustheincreaseinbedcapacityminusthedecreaseinbedcapacity.Constraint( 2{5 )isthebudgetconstraint 18 PAGE 19 2{6 )ensuresthatthenumberofbedsavailableandchangesinbedcapacityareintegervalued. 7 ].Inpractice,bedcapacityisincreasedordecreasedinbatches,andistypicallychangedinintegermultiplesofabasevalue,say,inmultiplesof10or25correspondingtothesizeofaunit.Asaresult,thereareonlyalimitednumberofchoicesforchangingcapacityineachperiod.Therefore,constraintsthatcapturethechangeincapacitycanbereplacedbyasetofdiscretealternativeconstraints,requiringthatonlyonealternativeischoseninthesolutionforeachperiod.Then,theoriginalnonlinearintegerprogrammingproblembecomesanonlinearbinary(i.e.,zeroone)integerprogrammingproblem,whichwerefertoastherestrictedbedcapacityplanning(RBCP)problem. IntheRBCPproblem,wearegivenabasevalueofBinmultiplesofwhichthebedcapacitycanbeincreasedordecreasedandweletnbethenumberofpossibledistinctlevelsofcapacityincreaseordecrease.Thatis,givenbedcapacitycinperiodt,thebedcapacityinperiodt+1canbeoneof(cnB)+,(c(n1)B)+,...,(cB)+,c,c+B,...,c+(n1)B,c+nB,where(x)+=maxf0;xg.Weassumethatallacquirednewadditionalcapacityisavailableandbecomeseectivecapacityinthesameperiod.Letz+it=1iftheavailablebedcapacityisincreasedbyiBatthebeginningofperiodtfori=1,2,...,n;and0otherwise.Similarly,letzit=1ifthebedcapacityisdecreasedbyiBatthebeginningofperiodtfori=1,2,...,n;and0otherwise.WecannowformulatetheRBCPproblemasanonlinearzerooneintegerprogrammingproblemasfollows: minTXt=1f(xt;t;t)+TXt=1g(xt1;xt)+TXt=1h(xt) (2{7)subjectto 19 PAGE 20 AsintheAHBCPproblem,objectivefunction( 2{7 )minimizesthetotalcostofpatientdelay,changingthebedcapacity,andoperatingtheexistingbedcapacity,constraint( 2{8 )imposesamaximumallowablelimitontheexpectedpatientdelay,constraint( 2{9 )setstheinitialbedcapacity,andconstraint( 2{10 )isaowbalanceequation.Constraint( 2{11 )ensuresthatonlyonechoiceforchangingthecapacityisallowedineachperiod.Constraint( 2{12 )imposesthebudgetconstraintontheamountofmoneyallocatedtochangingbedcapacity.Constraints( 2{13 )and( 2{14 )ensurethenonnegativityofthebedcapacitylevelandcapacitylevelselectiondecisionvariables,respectively. AnattractivefeatureoftheRBCPproblemisthatanetworkrepresentationcanbedeveloped.ConsideraTpartitegraphwithTlayerseachrepresentingatimeperiodt=1,2,...,Tintheplanninghorizon.Let(t;c)denotethesystemwhentherearecbedsinperiodt.LetC(c)bethesetofreachablecapacitylevelsinthenextperiodifthecapacityinthecurrentperiodisc,andwehaveC(c)=f(cnB)+;(c(n1)B)+;:::;(cB)+;c;c+B;:::;c+(n1)B;c+nBg.LetStbethesetofallcapacitylevelsreachableinperiodtfromallcapacitylevelsinperiodt1.Letdsbeasupercialsourcenodeconnectedonlytonode(0;x0)withzeroarclength.Node(0;x0)representsthebeginningstatewheretherearex0bedsinthehospitalattimet=0.Let(0;x0)beconnectedtoallnodes(1,x')forx'2C(x0).Ifw(x';1;1)1(i.e.,thepatientwaitingtimeconstraintisnotviolated)andg(x0;x')1(i.e.,thebudgetconstraintisnotviolated),thenthe 20 PAGE 21 Figure 21 providesanexampleofthenetworkrepresentationfortheRBCPproblemwherec0=300,B=25,n=1,andT=4.Inthisgure,apathfromthesupercialsourcenodetothesupercialsinknoderepresentsaplanforthebedcapacityovertheplanninghorizon.TheshortestpathwithoutcontaininganyarcwithcostMyieldsthecapacityplanwithtotalminimumcostthatobeysthepatientwaitingtimeandbudgetconstraintsovertheplanninghorizon.Ifnosuchpathcanbefound(i.e.,theshortestpathcontainsatleastonearcwithcostM),thentheproblemisinfeasibleandnocapacityplanthatobeysthewaitingtimeandbudgetconstraintsovertheplanninghorizoncanbefound. Figure21. NetworkowrepresentationforRBCPwithc0=300,B=25,n=1,andT=4 21 PAGE 22 2 ]. AswiththeRBCPproblem,anetworkrepresentationcanbedevelopedfortheRBCPwSproblem.ConsideraTpartitegraphwithTlayerseachrepresentingatime 22 PAGE 23 Figure 22 providesanexampleofthenetworkrepresentationfortheRBCPwSproblemwherec0=275,c0=300,B=25,n=1,andT=4.Foreaseofexposition,thethinarcsrepresentopening,maintaining,orshutteringofexistingcapacity,whereasthethickarcsrepresenttheacquisitionofnewcapacity.Inthisnetwork,apathfromthesupercialsourcenodetothesupercialsinknoderepresentsaplanforthebedcapacitythroughout 23 PAGE 24 Figure22. NetworkowrepresentationforRBCPwSwithc0=275,B=25,n=1,andT=4 RecallingtheRBCPproblem,wehavespeciedthenumberofarcsandnodesinthenetworktodeterminethetimetoobtaintheoptimalsolution.However,fortheRBCPwSproblem,sinceexistingcapacitycanbeincreasedfurtherthroughcapitalacquisition,theanalysisbecomesslightlymorecomplicatedanddependentontheinitialstate(i.e.,theamountofeectiveandexistingbedcapacity).Ifnoadditionalcapacity 24 PAGE 25 2 ]. Thisfacilitywouldliketodetermineanoptimalbedcapacityplanforthenexteightquarters,correspondingtoitsoperational,budgetary,andstrategicplanningperiods.Becausecapacityplanningmayinvolveasubstantialcapitalcommitment,itisimperativethatanycapacityexpansionplanbecarefullydevelopedandjustiedbaseduponthe 25 PAGE 26 2.4.1 ,andwerefertoitasS1.Table 21 liststherelevantparametersettingsforS1,andotherexperimentalscenariosrelativetothisscenarioaregiveninTable 22 Attheoutset,weprovideanestimatedrangeofdemandforthefacilityovertheplanninghorizon.Normally,asingleseasonallyadjustedtrendlinewouldbecomputedtoforecastthepatientarrivalratebasedonhistoricdemanddata.Instead,toillustratetheextentofvariationindemand,Figure 23 displaysasetofsimulatedpatientarrivalratesovertheplanninghorizonbaseduponthescenariosgiveninTable 22 .Wenotethatsomeoftheparameterchangesdirectlyimpactthepatientarrivalrate,anddierentpatientarrivalratesaregenerated.InS1,S3,S4,S5,andS6,thechangedparametersdonotimpactthearrivalratefunction,sothesescenarioshaveidenticalarrivalrates.(Note 26 PAGE 27 Parametersettingsforthebasescenario,S1 ParameterValue LengthoftheplanninghorizonT=8quarters,t=1,2,...,8 Forecasteddemandpertimeperiodt Initialexistingbedcapacityc0=350 Initialeectivebedcapacityc=350 Numberoflevelsofcapacityincreaseordecreasen=2 IncrementalamountofcapacitychangeB=10 Costtooperateaneectivebed$2,000/bed Costtoshutteraneectivebed$2,500/bed Costtoreactivateashutteredbed$2,500/bed Costtoacquireanewbed$200,000/bed (i.e.,expandcapacitythroughcapitalinvestment) Coecientofvariationforarrivalscat=0:5 Coecientofvariationforservicecst=0:5 Maximumexpecteddelayperpatientat=1hour Costofwaiting$300/hour Serviceratet=15:8patientsperbed Table22. Scenariodescriptionsforexperiment1 ScenarioDescriptionParameterchange S0Leveldemandb=0S1BasescenarioS2Increasedrateofdemandb=256S3Higherdemandvariabilitycat=2.0S4Higherservicevariabilitycst=2.0S5Highercostofwaitingperpatient$1,200/hourS6Smallermaximumexpecteddelayperpatientt=0.25ofanhour thatwithS3,highervariabilityinthearrivalrateimpactstheperformanceconstraintforaveragewaitingtime,notthearrivalratefunction.)ForS0andS2,thepatientarrivalratefunctionhasnotrendandahighertrendcomparedtoS1,respectively.Hence,arrivalratesgeneratedforthesescenariosaresignicantlydierentfromeachotherandS1. WehaveimplementedournetworkowapproachusingtheC++programminglanguageandsolvedforthescenariosusingapersonalcomputerwith3.0GHzPentiumIV 27 PAGE 28 Patientarrivalrateforexperiment1 processorand512MBRAMmemory.WeobtainedtheoptimalsolutionforeachscenarioandtheresultsaredepictedinFigure 24 ,whereeachlinerepresentstheoptimalcapacityplanthatcorrespondstooneofthesevenscenarios. Figure24. Optimalcapacityplansforexperiment1 28 PAGE 29 24 ,wehavethefollowingobservations.ForS1,werstobserveageneralreductioninthebedcapacity,thenagradualincreaseneartheendoftheplanninghorizon.Theinitialbedcapacityseemstobehigherthanneeded,andasaresult,thebedcapacityisreducedtoreducetotalcostsovertheplanninghorizonwhilemaintainingtheaveragewaitingtimeconstraint.Ofcourse,whenthedemandincreasesduetotheunderlyingtrend,thebedcapacityisincreased.WhendemandislevelasinS0,alowerenvelopeisformedrelativetothebasecase(i.e.,thebedcapacityforS0islessthanorequaltothebasecase).Similarly,withanincreasedrateofdemandasinS2,anupperenvelopeisformedrelativetothebasecase.WithincreasedvariationasinS3andS4,theoptimalcapacityplansaresimilartoS1'scapacityplanbuttendtorequirehighercapacitywhenthearrivalrateisincreasing.Whenthearrivalrateincreasesinperiods6,7,and8,becausethehigherarrivalvariabilityandhigherservicevariabilityaecttheaveragewaitingtimeconstraint,morecapacityisrequiredtokeepfromviolatingthisperformanceconstraint.Likewise,withahighercostofwaitingperpatientasinS5oratighteraveragewaitingtimeperformanceconstraintasinS6,theoptimalcapacityplanstendtorequirecapacityslightlyhigherthanthebasecase.Notsurprisingly,thenetresultofthisexperimentindicatesthatoptimalbedplansaredrivensubstantiallybychangesindemand.Whilehealthcaredecisionmakersmaynotbeabletoaectoveralldemandfortheirservices,iftheycanreducevariabilityinarrivals[ 78 ]orarewillingtotoleratealessstringentperformanceconstraint,lesscapacitywillberequired. 29 PAGE 30 Intherstpartofthisexperiment,wexthenumberoflevelstovarybedcapacityandthedurationoftheplanninghorizoninadditiontosomeotherproblemparametersconstantandexaminetheimpactofdierentratiosofexistingtoeectivebedcapacity.Usingtheassumptionsforthebasecasescenario,S1,fromthepreviousexperiment,weconsidertendierentlevelsoftheeectivebedcapacityintheinterval[260,350].Wegenerated30randomtestinstancesforeachoftheselevelsandthesummaryresultsareprovidedinFigure 25 andTable 23 Figure25. Numberofnodesinthenetworkasafunctionofinitialeectivebedcapacity InFigure 25 wedepictthenumberofnodesinthenetwork,andinTable 23 wereportthetimetobuildthenetworkandtimetoobtainthesolutionforeachleveloftheinitialeectivebedcapacity.Thenumberofnodesincreasesastheratioofeectivebedcapacitytoexistingbedcapacityapproachesone,andthisbehaviorisclearlydepictedinFigure 25 .However,inTable 23 ,weseethatanincreaseinthesizeofthenetwork 30 PAGE 31 Table23. SummarystatisticsfortheRBCPwSproblem'ssolutiontime(inCPUseconds)asafunctionofinitialeectivecapacity ofeectiveBuildthenetowrkObtainthesolution(inCPUseconds) bedcapacityMin.Avg.Max.Min.Avg.Max.Min.Avg.Max. 2600.00.00.20.00.00.00.00.00.22700.00.00.10.00.00.00.00.00.12800.00.00.10.00.00.00.00.00.12900.00.00.10.00.00.00.00.00.13000.00.00.10.00.00.00.00.00.13100.00.00.10.00.00.00.00.00.13200.00.00.10.00.00.00.00.00.13300.00.10.10.00.00.00.00.10.13400.10.10.10.00.00.00.10.10.13500.10.10.10.00.00.00.10.10.1 24 FromTable 24 ,asthenumberoflevelsofcapacitychangeandthelengthoftheplanninghorizonincreases,thenumberofnodesinthenetworkincreases.Theincreaseinthenumberofnodesimpactsthetotaltimerequiredtoobtaintheoptimalsolution.However,acloserexaminationoftheresultsrevealstheincreaseinthenumberofnodesinthenetworkhasadirectimpactonthetimerequiredtobuildthenetwork,andhasalmostnoimpactonthetimetoobtainthesolution.Onlyinthesettingwiththelargesttestinstances(i.e.,n=5andT=16)doweobserveanincreaseinthetimetoobtainthe 31 PAGE 32 SummarystatisticsfortheRBCPwSproblem'ssolutionasafunctionofcapacitylevelsandthelengthoftheplanninghorizon NodesinthenetworkArcsinthenetworkBuildthenetworkObtainthesolutionAvg.(Min.,Max.)Avg.(Min.,Max.)Avg.(Min.,Max.)Avg.(Min.,Max.)Avg.(Min.,Max.) (2,8)323.7(194,490)1,213.9(737,1,865)0.04(0.02,0.14)0(0,0)0.04(0.02,0.14)(3,8)707.1(451,982)3,566.3(2,226,5,061)0.13(0.06,0.2)0(0,0.02)0.13(0.06,0.2)(4,8)1,257.5(858,1,642)7,989.3(5,273,10,665)0.31(0.17,0.44)0(0,0.02)0.31(0.17,0.44)(5,8)1,980(1,455,2,470)15,199(10,834,19,349)0.64(0.41,0.89)0(0,0.02)0.64(0.41,0.89)(2,12)997.7(662,1,338)4,067.3(2,661,5,521)0.19(0.09,0.34)0(0,0.02)0.19(0.09,0.34)(3,12)2,296.2(1,697,2,858)12,850.4(9,328,16,195)0.63(0.38,0.86)0(0,0.02)0.63(0.39,0.86)(4,12)4,165.3(3,278,4,950)29,679(22,957,35,669)1.72(1.16,2.25)0(0,0.02)1.72(1.16,2.25)(5,12)6,607.5(5,469,7,614)57,209.8(46,688,66,583)4.17(3.02,5.31)0(0,0.02)4.17(3.02,5.31)(2,16)2,468.2(1,574,3,282)10,586.7(6,637,14,225)0.62(0.31,0.99)0(0,0.02)0.62(0.31,0.99)(3,16)5,659.9(4,110,6,954)33,550.5(23,969,41,609)2.25(1.36,3.36)0(0,0.02)2.25(1.36,3.36)(4,16)10,211.9(8,022,11,986)77,310.6(59,925,91,473)6.83(4.83,8.84)0(0,0.02)6.83(4.83,8.84)(5,16)16,128(13,278,18,378)148,628.7(120,987,170,537)57.57(13.13,138.22)0.49(0,1.25)58.06(13.13,139.28) PAGE 33 Inourmodel,wetreattheperformanceconstraintsasahardconstraint.Thatis,ifaparticularcapacitylevelviolatestheperformanceconstraint,thenasolutionwiththatparticularcapacitylevelisnotfeasible,andisdroppedfromfurtherconsideration.However,theperformanceconstraintcanbemodeledasasoftconstraintwherewecandeliberatelyallowtheviolationoftheperformanceconstraintwhileincurringapenaltycosttobeaddedtotheobjectivefunction.Wecanjustifythisconstraintbynotingthatlagstypicallyexistbetweencapacitylevelssotheremightbeperiodsoftimewherethefacilityisoperatingaboveitstypicalutilizationandthecapacityexpansioncannotoccurquicklyenoughtoallowtheorganizationtoreacttothechangeindemand.Toillustratehowourmodelcanbereformulatedwiththesoftconstraint,letvtbetheamountoftheviolation,stbetheamountofslackintheperformanceconstraint,and(vt)bethepenaltycostincurredforviolatingtheperformanceconstraintinperiodt.Then,consideringtheRBCPproblem,objectivefunction( 2{7 )wouldbereplacedwith:minTXt=1f(xt;t;t)+TXt=1g(xt1;xt)+TXt=1h(xt)+TXt=1(vt) (2{15) Similarly,constraints( 2{8 )and( 2{13 )wouldbereplacedwith: 33 PAGE 34 2{17 )and( 2{18 ),respectively,oncew(xt;t;t)isknown.Theonlymodicationofthenetworkistoincludethecostassociatedwithviolatingtheperformanceconstraint. Whenevaluatingahospital,werecognizethattheaveragewaitingtimetobeassignedabedorhavingexpenseswithinbudgetarenottheonlymetricstoassessfacilityperformance.Indeed,itmaybenecessarytoincludemeasuresforfacilityutilization,likelihoodofpatientdiversion,andthelike.Regardless,wenotethatmoreperformanceconstraintscaneasilybeaddedtotheformulationsfortheBCP,RBCP,andRBCPwSproblems.Anincreaseinthenumberofperformanceconstraintsdoesnotincreasethetimetoobtainthesolutionsignicantly,asthereisonlyaneedtotaketheseadditionalconstraintsintoaccountinsettingupthenetworkandassigningalargearccostincaseanyoftheconstraintsareviolated.Therefore,ourmodelingapproachisrobustandadditionalconstraintscanbeconsideredwithoutincreasingthecomplexityoftheformulationsignicantly. 34 PAGE 35 Ourmodelisbasedonagenericviewofahospitalwherewehaveassumedthatthedemand(i.e.,patientarrivals)andservice(i.e.,beds)componentsarehomogeneous.Fromanaggregateplanningperspective,suchuniformitymaybeacceptable.However,inordertoapplythisresearchtooperationaldecisionsupportforhealthcaredelivery,thereareadditionalavenuesofresearchworthpursuing.First,ifcostdependsonallpreviousstages,forexample,thecostofmaintainingthebedsdependsnotonlyonthenumberofbedsbutalsothedurationofthebedsareinthesystem,thenthenumberofverticesinthenetworkwillbeexponentialwithrespecttoTandtheoptimalsolutiontothenetworkwillnotbesolvedwithapolynomialtimealgorithm.Consequently,alternativemodelformulationsandsolutiontechniquestodeterminetheoptimalbedplanwouldbenecessary.Second,recognizingthathospitalbedsarenotidentical,facilitycapacitycouldbeseparatedtodistinguishthevariousspecialties,withspecialtyspecicdemandrates,lengthsofstay,andcosts.Indeterminingtheaveragewaitingtimeassociatedwithbeingassignedabed,wehaveusedclosedformapproximationstocalculatethisstatistic.Therefore,weareimplicitlyassumingthatthisgeneraldistributionaccountsfordierenttypesofpatientsthatrequiredierenttypesofhospitalbasedhealthcare.Thismaynotnecessarilybethecase,andshouldbeinvestigatedfurther.Third,ourworkcanbeexpandedtoincludemultipletypesofpatients(e.g.,electives,admissionscomingthroughtheemergencydepartment,andreferralsfromphysicians).Also,inestimatingthecostofpatientwaiting,weassumethatthiscostisidenticalregardlessofpatienttype.Clearly,forexample,thereshouldbedierentwaitingcostsassociatedplacinganemergencydepartmentadmissioninanappropriateunitversusaninappropriateunit.Assuch,representationsofpatientwaitingcostneedtobedevelopedinthepresenceofcongested,heterogeneousresources.Fourth,thecurrentformofourmodeldoesnotaccountforthepotentialtimedelaythatmayexistbetweenthedecisiontoexpand 35 PAGE 36 3 ]andtheongoingdebateregardingnursetopatientratios[ 4 ],theabilitytousephysicalcapacityhingesupontheavailabilityofsuitablemedicalpersonnel.Anaturalextensionofourmodelwouldbetoincorporateworkforceplanningtosimultaneouslydeterminethequantityandcompositionofthehealthcareresourcestoconstructacomprehensivecapacityplan. 36 PAGE 37 102 ],thehealthcareindustryaccountsforthelargestsectoroftheeconomyintheUnitedStates(US).Despiteadvancesinmedicaltechnologyand,thereby,theincreasinguseofmedicaldiagnostic,monitoring,andtreatmentequipment,thehealthcareindustryishighlylaborintensive.AccordingtotheUSDepartmentofLabor,thehealthcareindustryprovided13.5millionjobsin2004,outofwhich13.1millionjobsareforwageandsalaryworkersandabout411,000arefortheselfemployed[ 107 ].Itfollowsthatpersonnelwagesandsalariesaccountforthelargestportionofthetotalexpendituresforanyhealthcarefacility.Forinstance,hospitalsspendonaverageabout54percentofallexpendituresonwagesandsalaries[ 86 ].Hence,healthcarepersonnelplanning,i.e.,determiningtheappropriatemixofhealthcarepersonnel,neededtoprovidesafe,eective,timely,andcostecientservicetopatients[ 55 ],isanimportantproblem. Inpractice,bothininpatientandoutpatientfacilities,ahealthcareteam,comprisedofagroupofhealthcarepersonnelwithdierent,andcomplementary,skillsets,provideshealthcareservicestoindividualpatients[ 34 ].Themembersoftheteam(e.g.physicians,physician'sassistants,andregisterednurses)areresponsibletoperformasetoftasksrequiredforthediagnosis,monitoring,andtreatmentofthepatients.Someadditionaltasksmayhavetobeperformedbyotherpersonnel(e.g.,laboratorytechnicians,radiologicaltechnicians,andradiologists)whoarenotapartofthehealthcareteambutprovideassistancefordiagnosisandtreatment.Inthedeliveryofservicesbyhealthcareteams,thesafetyandeectivenessoftheserviceisensuredbytheappropriateselectionoftheservicecapabilityoftheteam,whereasthetimelinessandcosteectivenessoftheserviceisensuredbytheappropriateselectionoftheservicecapacityoftheteam.Theservicecapabilityofateamischaracterizedbythecollectionoftheskillspossessed 37 PAGE 38 Inahealthcarefacility,therearetypicallymultipletypesofhealthcareteamseachwithdierentcapabilities.Thepatientsthatarrivetothefacilityareclassiedaccordingtotheirconditions(i.e.,acuitylevels)ormedicalrequests.Basedonthisclassication,eachpatientisassignedtoahealthcareteamandadmittedtoanexamination/treatment(E/T)roomthathastheequipmentnecessarytoprovidetheserviceneededbythepatient.AsthesetoftasksaresharedamongthemembersoftheteamandatypicalfacilityhasmultipleE/Trooms,ahealthcareteamusuallyservesmultiplepatientssimultaneously.Forinstance,whiletheteamwaitsforthetestresultsfromthelabforapatient,aregisterednursemaybecollectingspecimensfromanother,aphysician'sassistantmaybesuturingawoundofanother,andaphysicianaccompaniedbyaregisterednursemaybediscussingatreatmentplanwithanother.Inourwork,weconsidertwoparticularsettingswherehealthcareservicesareprovidedbyteams. ShandsatAlachuaGeneralHospitalinGainesville,Floridaisacommunityhospitalthatprovidesemergencymedicineservices.triagenurse,whodeterminestheacuitylevelofthepatientandidentieswhetherthepatientrequiresimmediate(i.e.,emergency)ordelayed(i.e.,urgent)care.Theemergencycareservicesaredeliveredbyanemergencycare(EC)team,whichiscomposedofphysiciansandregisterednurses,andtheurgentcareservicesaredeliveredbyanurgentcare(UC)team,whichisalsocomposedofphysiciansandregisterednurses.Wenotethatthetriagenursedoesnotbelongtoeitheroftheseteams,butactsasagatekeepertorouteanarrivingpatienttoeitheroftheteams.ThereareeightandtwoE/TroomsdedicatedtotheECandUCteams,respectively.Moreover,theECteamhasthecapabilitytoattendtourgentcarepatients,buttheUCteamdoesnothavethecapabilitytoattendtoemergencycarepatients.Finally,emergencycarepatientshavepreemptivepriorityoverurgentcarepatients.Thatis,ifanemergencycarepatientarrivestotheED,whileallE/TroomsdedicatedtotheEC 38 PAGE 39 TheWomen'sClinicattheStudentHealthCareCenterattheUniversityofFloridainGainesville,Floridaprovideswomen'shealthcareservices.Theoutpatientclinic(OC)servesnotonlynonacutepatients,i.e.,thosewhoneedroutineservices,buttreatsacutepatientsalso.Theservicesforthediagnosisandtreatmentofacuteillnessesandabnormalitiesareprovidedbyaphysician(P)team,whichiscomposedofaphysicianandaphysician'sassistant.Theroutineclinicalservicesaredeliveredbyanursepractitioner(NP)team,whichiscomposedofanursepractitionerandaregisterednurse.TherearefourE/TroomsdedicatedtotheNPteamandtworoomstothePteam.Moreover,thePteamhasthecapabilitytoattendtononacutepatientsbuttheNPteamdoesnothavethecapabilitytoattendtoacutepatients.Finally,acutepatientshavenonpreemptivepriorityoverthenonacutepatients.Thatis,ifanacutepatientarrivestotheOC,whileallE/TroomsdedicatedtothePteamareoccupiedbyotherpatientsandoneofthemisanonacutepatient,thenthePteamdoesnotinterruptservicetothenonacutepatientandtheacutepatienthastowaituntilanE/TroomdedicatedtothePteambecomesavailable. Inthesettingswediscussedabove,theservicecapabilitiesofthehealthcareteamsarexedasdictatedbytheserviceneedsofthepatients,andpersonnelplanningismainlyconcernedwithdeterminingtheservicecapacityoftheteams.Indeterminingtheservicecapacity,however,administratorsmusttakeseveraladditionalfacilitycapacity,budgetary,andlegislativeconstraintsintoaccountthatlimittheminimumandmaximumtotalnumberofeachpersonneltypeemployed.Forinstance,budgetconstraintsmaylimitthetotalnumberofphysiciansemployed,whereaslegislatednursetopatientstangratiosmayprescrobealowerlimitonthetotalnumberofregisterednurses.Therefore,giventhelowerandupperboundsonthetotalnumberofpersonnelwithdierentskillsthatcanbe 39 PAGE 40 Inthispaper,motivatedbythepracticalsettingswediscussedabove,weaddressthelongtermhealthcareteamservicecapacityplanning(HCTSCP)probleminthecontextofhealthcarefacilitieswheretherearetwopatientclassesandtwotypesofteams.Weassumethatthecapabilityofeachtypeofteamisknown,andthenumberofE/Troomsallocatedtoeachtypeofteamisxedovertheplanninghorizon.Therefore,theservicecapacityofateamcanonlybechangedbymodifyingthecongurationoftheteam.Moreover,theservicecapacityofateamcanbequantiedbytheservicerateoftheE/Troomsdedicatedtotheteam,whichisthereciprocaloftheaveragetimethatthepatientsspendintheE/Troomspriortodischargeortransfertoanotherdepartment.Weformulateanonlinearbinaryintegerprogrammingmodeltodeterminetheservicecapacityplanforthehealthcareteamssuchthathealthcareservicesaredeliveredinatimely(i.e.,theaveragetimeapatientfromaparticularclassspendsinthesystemdoesnotexceedaprespeciedthreshold)andcosteective(i.e.,thetotalcostsassociatedwithchangingservicecapacitybyhiringadditionalpersonnel,reassigningexistingpersonnelorlayingoexistingpersonnelandoperatingtheservicecapacityareminimized)manneroveragivenplanninghorizonconsideringsomeadditionalconstraints. Toestimatetheaveragetimeapatientfromaparticularclassspendsinthesystem,wedevelopqueuingmodelsanddecompositionbasedapproximation(DBA)methods. 40 PAGE 41 Theremainderofthispaperisorganizedasfollows.InSection 3.2 ,wereviewtherelatedliterature.Section 3.3 presentsourcapacityplanningmodelforHCTSCPandshowhowitcanbeinterpretedasanetworkowmodel.WedevelopqueueingmodelsfortheEDandOCsettingsandpresentDBAmethodstoanalyzethesemodelsinSection 3.4 .InSection 3.5 ,wepresentresultsfromourcomputationalstudythatevaluatestheaccuracyoftheapproximationsfromSection 3.4 andinvestigatethecomputationalperformanceoftheformulationpresentedinSection 3.3 .Section 3.6 includesadiscussionoftheresultsandsuggestsfutureresearchdirections. 41 PAGE 42 61 ].Burkeetal.[ 28 ]provideanexcellentreviewoftheexistingworkinthisarea. Analyticalworkthatfocusesonmediumtolongtermpersonnelplanningisrelativelylimitedinscope.SchneiderandKilpatrick[ 97 ]developoptimizationmodelsforpersonnelplanninginhealthcarefacilities.KaoandTung[ 62 ]presentalinearprogrammingmodelfortheaggregate(nursing)workforceplanningproblem,whichislaterextendedbyBruscoandShowalter[ 27 ]toaccountfortheexogenousimpactofnursingshortage.KroppandCarlson[ 72 ]proposetheintegrateduseofoptimizationandsimulationmodeling.ExistingworkthatusessimulationmodelingisreviewedinJunetal.[ 58 ].Althoughtheearlierworkinthisareaprimarilyfocusesontheperspectiveofthehealthcareproviderbyplacinganemphasisonminimizingthecostofpersonnelresourcesormaximizingtheutilizationofpersonnelresources,thereisaneedtotakethepatient'sperspectiveintoaccountbyconsideringthetimeliness(e.g.,minimizingofthewaitingtimeand/ortimeinsystem)ortheavailability(e.g.,minimizingtheprobabilityofndingallserversbusyandbeingdivertedtoanotherserviceprovider).Tothisend,wefocusonminimizingthesumofcapacitycostsandcostofnotprovidingserviceinatimelymanner,whileensuringthattheaveragetimeinsystemforapatientdoesnotexceedaprespeciedthreshold.Tothisend,wedividetheplanninghorizonintodiscretetimeperiodsofequallengthandassumethatthesystemachievessteadystateineachoftheseintervals.Thisallowsustousequeueinganalysistocapturethestochasticbehaviorofthesystemandcomputethe 42 PAGE 43 5 ]. Intheliterature,thereisanumberofstudiesthatinvestigatequeueingsystemsthatarecloselyrelatedtothesettingweconsider.StanfordandGrassmann[ 105 ]derivetheexpectedwaitingtimeinacallcenterwithunilingualandbilingualserversservingmajorityandminoritylanguageusecustomers.Aminoritylanguageusecustomercanonlybeservedbyabilingualserver,andthetypeofacustomerisnotbeknownpriortotherstservice.Theserviceratesforallservertypesarethesame,i.e.,independentofthecustomertype.Green[ 44 ]andShumsky[ 99 ]alsoinvestigateexpectedwaitingtimeofasystemwithtwotypesofcustomersandlimitedandgeneraluseservers.Bothofthemconsiderthecasewheretheserviceratesdependonthetypeoftheserver.Hence,thedistinguishingcharacteristicofthesystemweinvestigateisthedependenceoftheservicetimeoncustomerclassforoneoftheservertypes.Thisseeminglyasimpleattributeofthesystemleadstosignicantmodelingchallenges. Inqueueingtheory,toobtainthestationaryprobabilitiesofasystem,themajorapproachesusedincludethematrixanalyticmethods[ 44 60 74 75 87 105 ],powerseriesalgorithm(PSA)approach[ 19 20 54 71 ],andDBAmethod[ 99 ].ThematrixanalyticmethodsformulatethesystemasaMarkovchaintowhichthestationaryprobabilityhasthematrixgeometricformn=0Rn,whereratematrixRcanbeobtainedthroughaniterativealgorithm[ 87 ].ThePSAapproachrstrepresentsthestationaryprobabilitiesofasystemaspowerseriesexpansionsofthetracintensityofthesystem,andthenrecursivelysolvesforthecoecientsofthepowerseriesexpansionsbyusingthesetofstationaryequations.KaoandWilson[ 64 ]comparetheperformanceofthePSAandmatrixanalyticmethodswiththreeiterativealgorithmsproposedin[ 60 74 75 ],andconcludethatthePSAperformsextremelywellintermsofcomputationalspeed,thoughitmayencounterdicultiesinparametersettingswhichmayleadtolossesinaccuracy.Shumsky[ 99 ]proposestheDBAmethod.TheDBAmethodrstdividesthestatespace 43 PAGE 44 99 ]illustratesthatthismethodgeneratesperformancemeasuresrapidlywithsucientaccuracythatcanbeusedincallcentercapacitydecisions.Inourqueueinganalysis,wealsousethisDBAmethod. Therearetwotypesofhealthcareteams,indexedbyj=1;2,andtwotypesofE/Trooms.Teamtype1canprovideservicetobothpatientclasseswithdierentserviceratesinitsdedicatedE/Trooms,type1,whileteamtype2canonlyprovideservicetopatientclass2initsdedicatedE/Trooms,type2.LetrjdenotethenumberofE/Troomsallocatedtoteamtypejforj=1;2,andrbetheassociatedtwodimensionalvectorrepresentingtheroomallocation.Aswediscussedearlier,theservicecapacityofeachtypeofteamcanonlybechangedbymodifyingthecongurationoftheteamandcanbequantiedbytheserviceratesoftheassociatedE/Trooms.Giventhelowerandupperboundsonthenumberofpersonnelwithdierentskillsthatcanbeemployedinthefacilityandthetypesofpersonnelthatmustbeincludedinaparticulartypeofteam,letKjdenotethetotalnumberofpossiblecongurationsforteamtypej,indexedbyk=1;:::;Kj.Supposethatitispossiblefortheadministratorstoestimatethe 44 PAGE 45 Wehavetwosetsofdecisionvariables.Letxjkttakethevalueofoneifforteamtypejcongurationkisselectedinperiodtforj=1;2,k=1;:::;Kjandt=1;:::;T,andzerootherwise.Also,letxjtbeaKjdimensionalvectorassociatedwiththeteamcongurationdecisionforteamtypejinperiodt.Finally,letijtdenotetheservicerateforclassipatientswhoaretreatedintypejE/Troomsinperiodtfori=1;2,j=1;2andt=1;:::;T,andtbea22matrixassociatedwiththeserviceratesoftheE/Troomsinperiodt. Weletwi(t;t;b;r)andsi(t;t;b;r)representthewaitingcostofclassipatientsandtheaveragetimespentinthesystembyclassipatients,respectively,asafunctionoftheservicerateoftheE/Troomstinperiodt,patientarrivalratestinperiodt,maximumnumberofpatientsallowedinthefacilitybandE/Troomallocationr.Also,weletcj(xjt1;xjt)denotethecostofmodifyingthecongurationoftypejteamfromperiodt1totandoj(xjt)denotethecostofemployingthepersonnelnecessaryforthechosencongurationforteamtypejinperiodt.Assumingthatallacquiredadditionalpersonnelcapacityisavailableandbecomeseectiveinthesameperiod,theHCTSCPproblemcanbeformulatedasanonlinearbinaryintegerprogrammingproblemasfollows: min2Xi=1TXt=1wi(t;t;b;r)+2Xj=1TXt=1cj(xjt1;xjt)+2Xj=1TXt=1oj(xjt) (3{1)subjecttoKjXk=1xjkt=18j;t 45 PAGE 46 Theobjectivefunction( 3{1 )minimizesthesumofthecostassociatedwiththetimethatthepatientsspendinthesystem,thecostofmodifyingteamcongurationstochangetheservicecapacityoftheteams,i.e.,serviceratesoftheassociatedE/Trooms,andthecostofemployingthepersonnelnecessaryfortheselectedteamconguration.Constraints( 3{2 )stipulatethatonecongurationmustbeselectedforeachteamtypeineachplanningperiod.Constraints( 3{3 )assigntheserviceratesforE/Troomsaccordingtotheselectedcongurationsforeachteamtypewithrespecttodierentpatientclasses.Notethat12t=0fort=1;:::;Tdueto12k=0fork=1;:::;K2.Constraints( 3{4 )imposeupperlimitsontheaveragetimethatpatientsspendinthesystemforeachpatientclass.Constraints( 3{5 )limitstheamountoffundsthatcanbeallocatedtochangingteamcongurationineachplanningperiod.Finally,constraints( 3{6 )and( 3{7 )ensuretheintegralityofteamcongurationandnonnegativityofserviceratedecisionvariables,respectively. HCTSCPisadicultnonlinearbinaryintegerprogrammingproblemwithnonlinearconstraints.Notethat,however,HCTSCPcanberepresentedbyaT+1partitenetwork,whereeachlayerinthenetworkrepresentsatimeperiodt=0;:::;Tintheplanninghorizon.Let(k1;k2;t)denotethefacilitywhencongurationkjfortypejteamisusedinperiodt.Layert=0includeasinglenode(k1;k2;0),whichdenotestheinitialcongurationsfortype1andtype2teams.AsupercialsourcenodeSisconnectedtonode(k1;k2;0)onlywithzeroarccost.Eachlayert=1;:::;TcontainsK1K2nodes,eachofwhichrepresentsafeasiblepairofteamcongurationsforthetwoteamsinperiodt.Thecostofthearcsconnectinganode(k1;k2;t1)inperiodt1toanode(k1';k2';t)inperiodtisgivenbyP2i=1wi(t;t;b;r)+P2j=1cj(xjt1;xjt)+P2j=1oj(xjt),wherexjkj't=1forj=1;2,fort=1;:::;T.However,foragivennode(k1;k2;t),if 46 PAGE 47 3{4 )orconstraints( 3{5 )isviolated,thenthecostoftheincomingarcstothisnodearesettoM,whereMisaverylargenumber.Finally,eachnodeinlayert=TisconnectedtoasupercialsinknodeDonlywithzeroarccost.Figure 31 providesanexampleofthenetworkrepresentationforHCTSCPforK1=3,K2=3,andT=2.Foreachteamtype(a1;a2)representsacongurationwherethenumberoftype`personnelintheteamisa`for`=1;2.Inthisgure,apathfromthesupercialsourcenodetoSthesupercialsinknodeDrepresentsateamcapacityplanovertheplanninghorizon.TheHCTSCPproblemndsacapacityplanwithminimumcost,iftheshortestpathonthisgraphdoesnotcontainanyarcwithcostM.Otherwise,theproblemisinfeasible.Inthenextsection,weexplainhowweobtaintheaveragetimeinsystemforeachpatientclass. Figure31. AnillustrationofthenetworkrepresentationforHCTSCP 47 PAGE 48 Thetransitionrateoutofstate(n1;n2;m)2Spisthesumofarrivalanddepartureratesofbothpatientclasses,thatis,1I(n1 PAGE 49 Thepossibletransitionsenterstate(n1;n2;m)fortheEDsetting EventTransitionprobability 1)Aclass1patientarrivestothesystem 2)Aclass1patientarrivestothesystemandpreemptsaclass2patientservedinatype1E/Troom 3)Aclass2patientarrivestothesystem,andisadmittedimmediatelytoatype2E/Troomorjoinsthequeueofclass2patients 4)Aclass2patientarrivestothesystem,andisadmittedimmediatelytoatype1E/Troom 5)Aclass1patientdepartureresultsineitherfreeingatype1E/Troomorstartingtheserviceofawaitingclass1patient [(n1+1)^r1]11P(n1+1;n2;m)I(n1 PAGE 50 50 PAGE 51 Thetransitionrateoutofstate(n1;n2;m)2Snpisthesumofarrivalanddepartureratesofbothpatientclasses,thatis,1I(n1 PAGE 52 TwodimensionalCTMCapproximation twoindependentsimplesubsystemsandusetheperformancemeasuresofthesesimpliedsubsystemstoadjusttheresultsobtainedinstep(3). 32 .Subsystem1containsthetype1team,type1E/Troomsandthepatientsthatareservedinthetype1E/Trooms.Similarly,subsystem2containsthetype2team,type2E/Troomsandthepatientsthatareservedinthetype2E/Trooms.WethenuseatwodimensionalCTMCwithstate(z;n2')toapproximatetheoriginalthreedimensionalCTMC,wherezisthetotalnumberofpatientsinsubsystem1andn2'isthenumberofclass2patientsinsubsystem2.Notethatzincludesthenumberofclass1andclass2patientsinsubsystem1,andn2'includesthenumberofclass2patientsservedinsubsystem2only.ThestatespaceofthetwodimensionalCTMCisdenedbyS'np=f(z;n2'):0zr1+b1;0n2'r2I(z PAGE 53 wherePc2=[2P(N2'=r2jZr1)]=[1+2P(N2'=r2jZr1)]:NotethattheprobabilityP(N2'=r2jZr1)canbeapproximatedbyP(N2'=r2jZ PAGE 54 z=(r1E[MjZ=z])11+E[MjZ=z]21; wherethevalueofE[MjZ=z]followsfromequation( 3{9 ). 3{10 )andPon1istheprobabilitythatatype1serverisavailableforawaitingclass2patient,andcanbeapproximatedbyPon1=P(Z=r1jN2'>r2)(r1+2)=(r1+2+1). z=(min(z;r1)E[MjZ=z])11+E[MjZ=z]21; wherethevalueofE[MjZ=z]iscomputedfromequation( 3{8 )for0zr1,andfromequation( 3{9 )forr1 PAGE 55 1P(N2'>r2jZr1)P(Z PAGE 56 Notethattherstterminequation( 3{15 )representstheprobabilitythatanarrivingclass2patientndsalltype2E/Troomsoccupiedandoverowstosubsystem1immediately.Thesecondtermrepresentstheprobabilitythatanarrivingclass2patientwaitsinsubsystem2untilatype1E/Troombecomesavailable.LetE[ 3{12 )and( 3{13 )aredecreasedproportionally,i.e., 3{12 )and( 3{13 )areincreasedproportionally. 56 PAGE 57 106 ].AGEalgorithmavoidsunnecessaryrowoperationsbyconsideringthefactthatQisabandedmatrix,containingalargenumberofzeroelements.TheDBA,GE,andAGEmethodsareimplementedusingC++programminglanguage,andthenumericalresultsreportedareobtainedusingapersonalcomputerwitha3.0GHzPentiumIVprocessorand1GBRAMmemory. OurparameterchoicesforourcomputationalstudyarebasedonthedatacollectedfromaparticipatingED.Inthebasecase,thereareeightandtworoomsallocatedtotype1andtype2teams,respectively,i.e.,(r1;r2)=(8,2);serviceratesforclass1andclass2patientsintype1E/Troomare0.25and0.80patients/hour,respectively,i.e.,(11;21)=(0.25,0.80);servicerateforclass2patientsintype2E/Troomare0.75patients/hour,i.e.,22=0.75;andthemaximumnumberofpatientsfromeachclassallowedinsystemare20and20,respectively,i.e.,(b1;b2)=(20,20).Wenotethatinpractice,anEDisrequiredbylawtoadmitallthepatientsthatrequestemergencymedicineservices.Therefore,essentially,thewaitingroomcapacityisinnite,andnoarrivingpatientisdeniedofentrytothesystembecauseoflackofwaitingroomcapacity.However,whenthereisnotenoughservicecapacity,thenanarrivingpatientcanbedivertedtoasisterhospital.Therefore,formodelingpurposes,weincludealimitonthenumberofpatientsofeachtypeinthesystem. 57 PAGE 58 InTable 32 ,wereportthesizeoftheCTMCaswellastheCPUtime(inseconds)requiredtoobtainthesolutionusingtheDBA,AGE,andGEmethods.Inthepreemptivecase,thesizeoftheCTMCmodelgrowsifr1orr2increases.Inaddition,anincreaseinr1hasamoresignicanteectthanthatinr2.Similarbehaviorcanbeobservedinthenonpreemptivecase,however,theimpactofincreasingr1onproblemsizeinnonpreemptivecaseismoresignicantthanthatinthepreemptivecase.Furthermore,underthesameE/Troomallocation,theproblemsizeofnonpreemptivecaseisatleasttwicelargerthanthatofpreemptivecaseforthetestedscenariosinTable 32 .OurresultsshowthatthetimetoobtaintheapproximateresultsusingtheDBAmethodisnegligibleandAGEisconsiderablymoreeectivethanGEinobtainingtheexactsolution. Table32. ComputationalrequirementoftheDBA,AGE,andGEmethods PreemptiveNonPreemptive 224470.00.00.527070.00.310.8244530.00.00.627130.00.310.9284650.00.00.627250.00.311.2 424610.00.00.642250.00.736.6444810.00.00.742450.00.737.9485210.00.00.942850.00.739.8 825130.00.00.866090.01.9116.1825850.00.01.266810.02.0129.2827290.00.02.368250.02.1161.0 Unitofrunningtime:Second 58 PAGE 59 33 and 34 ,wereportthepercentageerrorassociatedwiththeexpectedtimeinsystemforthetwopatientclassesobtainedbytheDBAmethod(whencomparedtotheexactsolutionobtainedbytheAGEmethod)inthepreemptiveandnonpreemptivecase,respectively.Table 33 showsclearlythattheaveragetimeinsystemforclass1patientscanbecorrectlycomputedbytheDBAmethodbecausesubmodel1inSection 3.4.1 capturestheexactbehaviorofclass1patientsintheoriginalCTMC.Forclass2patients,therstexperimentshowsthatfortheinstanceswiththesamelevelsofr1and,theabsolutepercentageerror(APE)tendstodecreaseasr2increases.Forexample,fortheinstanceswithr1=2and=0.6,APEdecreasesasr2increases.Inaddition,forthesamelevelsofr1/r2and,APEtendstodecreaseasr1orr2increasesassystemutilizationislow,suchastheinstanceswithr1/r2=1and=0.6,APEdecreasesasr1(orr2)increases.Inotherwords,foragivensettingofr1orr1/r2and,DBAperformsbetterintheproblemswithlargerr2.Inthesecondexperiment,9scenariosofservicecapacityaretested.Forthesamelevelof(11;21),APEtendstoincreaseas22increasesfortheinstanceswith=0.6,whichisoppositetotheresultsoftheinstanceswith=0.9,whereAPEtendstodecreaseas22increases.Last,bothexperimentsshowthatDBAtendstounderestimateclass2patients'expectedtimeinthesystemwhenthesystemutilizationislow,i.e.,=0.6,whileoverestimateclass2patients'expectedtimeinthesystemwhenthesystemutilizationishigh,i.e.,=0.9. Table 34 showstheresultsfornonpreemptivecase.Intherstexperiment,weobservethatfortheinstanceswiththesamelevelsofr1and,theAPEforclass2patientstendstodecreaseasr2increases,whichisthesameasthepreemptivecase.Inthesecondexperiment,forthesamelevelsof(11;21)and,APEtendstoincreaseas22increasesforbothclass1andclass2patients.Inaddition,DBAtendstounderestimateclass1patients'expectedtimeinsystemwhileittendstooverestimateclass2patients'expectedtimeinsystem.Forclass1patients,APEislessthan5percentforalltestedinstances,i.e.,DBAyieldsamorereliableestimateforclass1patients'timeinsystem. 59 PAGE 60 Relativeandpercentageerror:preemptivecase 220.0(0.0%)5.3(6.1%)0.0(0.0%)5.3(4.4%)0.0(0.0%)0.1(0.0%)0.0(0.0%)10.1(3.1%)40.0(0.0%)5.2(3.4%)0.0(0.0%)8.2(4.1%)0.0(0.0%)10.0(3.6%)0.0(0.0%)9.5(2.3%)80.0(0.0%)3.1(1.1%)0.0(0.0%)6.6(1.8%)0.0(0.0%)10.8(2.4%)0.0(0.0%)12.5(2.2%) 420.0(0.0%)3.2(4.0%)0.0(0.0%)1.2(1.1%)0.0(0.0%)9.1(5.6%)0.0(0.0%)29.9(10.6%)40.0(0.0%)3.8(2.5%)0.0(0.0%)5.1(2.7%)0.0(0.0%)3.6(1.4%)0.0(0.0%)1.3(0.3%)80.0(0.0%)2.5(0.9%)0.0(0.0%)5.3(1.5%)0.0(0.0%)8.5(2.0%)0.0(0.0%)10.1(1.8%) 820.0(0.0%)2.3(3.1%)0.0(0.0%)0.6(0.6%)0.0(0.0%)14.9(11.0%)0.0(0.0%)51.5(22.9%)40.0(0.0%)2.7(1.9%)0.0(0.0%)2.9(1.6%)0.0(0.0%)2.2(0.9%)0.0(0.0%)15.4(4.5%)80.0(0.0%)1.9(0.7%)0.0(0.0%)3.9(1.1%)0.0(0.0%)5.6(1.3%)0.0(0.0%)5.9(1.1%) Average0.0(0.0%)3.3(2.6%)0.0(0.0%)4.4(2.1%)0.0(0.0%)7.2(3.1%)0.0(0.0%)16.2(5.4%) Experiment2:Eectsofservicecapacity,(r1;r2)=(8,2) (0.20,0.70)0.650.0(0.0%)2.4(3.2%)0.0(0.0%)0.3(0.4%)0.0(0.0%)14.1(10.3%)0.0(0.0%)49.3(21.7%)0.750.0(0.0%)3.6(4.7%)0.0(0.0%)1.6(1.6%)0.0(0.0%)10.5(7.3%)0.0(0.0%)41.4(17.5%)0.850.0(0.0%)4.9(6.2%)0.0(0.0%)3.5(3.4%)0.0(0.0%)7.1(4.8%)0.0(0.0%)34.6(14.1%) (0.25,0.80)0.650.0(0.0%)1.2(1.7%)0.0(0.0%)2.3(2.5%)0.0(0.0%)18.4(14.1%)0.0(0.0%)59.4(27.6%)0.750.0(0.0%)2.3(3.1%)0.0(0.0%)0.6(0.6%)0.0(0.0%)14.9(11.0%)0.0(0.0%)51.5(22.9%)0.850.0(0.0%)3.4(4.4%)0.0(0.0%)1.1(1.1%)0.0(0.0%)11.7(8.3%)0.0(0.0%)44.5(19.0%) (0.30,0.90)0.650.0(0.0%)0.4(0.6%)0.0(0.0%)3.9(4.3%)0.0(0.0%)22.0(17.5%)0.0(0.0%)68.0(33.2%)0.750.0(0.0%)1.3(1.8%)0.0(0.0%)2.3(2.5%)0.0(0.0%)18.6(14.2%)0.0(0.0%)60.0(27.9%)0.850.0(0.0%)2.3(3.0%)0.0(0.0%)0.8(0.8%)0.0(0.0%)15.5(11.4%)0.0(0.0%)53.0(23.7%) Average0.0(0.0%)2.4(3.2%)0.0(0.0%)1.8(1.9%)0.0(0.0%)14.8(11.0%)0.0(0.0%)51.3(23.1%) Unit:min.(%) PAGE 61 Relativeandpercentageerror:nonpreemptivecase 223.2(2.8%)1.1(1.3%)5.6(3.3%)4.5(3.7%)6.2(2.3%)14.8(8.1%)13.7(3.3%)23.6(7.4%)43.1(2.7%)2.3(1.5%)5.9(3.5%)2.6(1.3%)9.1(3.5%)10.4(3.8%)9.9(2.4%)14.2(3.4%)82.3(2.0%)3.8(1.3%)5.2(3.1%)0.2(0.1%)9.4(3.6%)4.5(1.0%)5.9(1.4%)4.6(0.8%) 422.7(1.6%)9.5(12.0%)5.5(2.4%)4.1(3.9%)9.5(2.9%)10.9(7.0%)12.9(2.7%)32.6(12.0%)42.8(1.6%)10.8(7.2%)6.1(2.6%)6.0(3.2%)10.9(3.3%)8.8(3.5%)15.0(3.1%)27.0(7.1%)82.1(1.2%)9.1(3.1%)5.6(2.4%)9.1(2.6%)11.2(3.4%)1.5(0.3%)16.0(3.3%)11.1(2.0%) 821.6(0.5%)19.6(26.6%)4.2(1.1%)19.0(20.4%)8.3(1.7%)8.5(6.6%)12.4(2.0%)13.0(6.1%)41.7(0.6%)21.4(14.7%)4.8(1.3%)22.5(12.7%)10.0(2.1%)11.0(4.8%)15.2(2.5%)13.7(4.1%)81.4(0.5%)11.5(4.0%)4.7(1.3%)25.2(7.3%)10.8(2.3%)17.5(4.2%)17.0(2.8%)1.4(0.3%) Average2.3(1.5%)9.9(8.0%)5.3(2.3%)10.3(6.1%)9.5(2.8%)9.8(4.4%)13.1(2.6%)15.7(4.8%) Experiment2:Eectsofservicecapacity,(r1;r2)=(8,2) (0.20,0.70)0.651.5(0.5%)20.7(28.1%)3.9(1.0%)19.8(21.2%)7.7(1.6%)7.4(5.7%)11.5(1.9%)19.5(9.1%)0.751.8(0.6%)22.7(29.7%)4.4(1.2%)21.7(22.3%)8.5(1.8%)8.1(5.9%)12.5(2.1%)22.5(10.0%)0.852.0(0.6%)24.4(31.0%)4.8(1.3%)23.4(23.2%)9.3(2.0%)8.8(6.2%)13.5(2.2%)25.1(10.8%) (0.25,0.80)0.651.4(0.5%)17.8(24.9%)3.7(1.0%)17.2(19.2%)7.4(1.6%)7.9(6.4%)11.3(1.9%)10.6(5.3%)0.751.6(0.5%)19.6(26.6%)4.2(1.1%)19.0(20.4%)8.3(1.7%)8.5(6.6%)12.4(2.0%)13.0(6.1%)0.851.8(0.6%)21.3(28.0%)4.6(1.2%)20.6(21.4%)9.1(1.9%)9.1(6.8%)13.4(2.2%)14.8(6.7%) (0.30,0.90)0.651.4(0.4%)15.4(22.1%)3.5(0.9%)15.1(17.4%)7.1(1.5%)7.9(6.7%)11.0(1.8%)4.0(2.1%)0.751.5(0.5%)17.1(23.8%)4.0(1.1%)16.7(18.6%)7.9(1.7%)8.6(6.9%)12.1(2.0%)6.2(3.1%)0.851.7(0.6%)18.7(25.3%)4.4(1.2%)18.3(19.6%)8.7(1.8%)9.1(7.1%)13.1(2.1%)8.2(3.9%) Average1.6(0.5%)19.8(26.6%)4.2(1.1%)19.1(20.4%)8.2(1.7%)8.4(6.5%)12.3(2.0%)13.8(6.4%) Unit:min.(%) PAGE 62 Weconsiderthepersonnelplanningproblemoverathreeyearplanninghorizonwheretheunitplanningperiodcorrespondstoaquarterofayear,i.e.,T=12.Weassumethat,forallt,theinitialserviceratesforclass1andclass2patientsare(110;210;220)=(0.25,0.80,0.75)patient/hour,theallowablemaximumnumbersofpatientsinsystemare(b1;b2)=(20,20)patients,theE/Troomallocationis(r1;r2)=(8,2),andtheupperboundontheamountoftimethatapatientspendsinthesystemare(1;2)=(4.75,4.00)hours.Weconsideracasewheretherearetwotypesofpersonnelincludedineachteam,andtherearesixandfourfeasiblecongurationsfortype1andtype2teams,respectively.Table 35 showsthenumberofdierenttypesofpersonnelwithdierentskillsetsincludedineachcongurationandthecorrespondingservicecapacityaswellasoperatingcostforeachteam.Withoutlossofgenerality,weassumethatpatientwaitingcostsincreaselinearlywithpatients'timeinsystem,unitdelaycostforeachpatientclass(UP1t;UP2t)aresetto($400,$100)/hourperpatient,andthefundsthatcanbeallocatedtochangetheservicecapacityoftheteamsinthefacilityt=$17,000forallt.Finally,weassumethatpersonnelhiringorterminationcostsarezero. AnexaminationofemergencymedicinepracticesshowthatpatientarrivalstotheEDexhibitseasonality.Wegeneratethetotalquarterlyarrivalrateofpatientsusingaseasonallyadjustedtrendline,representedbyafunctionoftheformt=mod(t;4)u(0+bt)whereiisthequarterlyseasonalityfactorforseasoni(wherewehavethefollowingestimatesfortheseasonalityfactors1=0.8,2=1.0,3=1.2,and4=1.0),uis 62 PAGE 63 Teamcongurations Type1TeamType2Team 1240.230.7556,0001110.6011,5002250.250.8063,0002120.7516,5003260.280.8570,0003221.0023,0004340.351.0070,0004231.1028,0005350.401.1077,0006360.451.2084,000 Unitofij:patients/hour.Unitofcost:$/quarter auniformlydistributedrandomnumber(wherewehaveuU[0:8;1:2]),0=2,b=0.04,andt=1;:::;12.Notethatweimplicitlyassumethatthepatientdemandincreaseslinearlyby2percenteveryquarter.Weassumethatthefractionofclass1patientisfit=0:85forallt,i.e.,1t=f1ttand2t=(1f1t)t.WenotethatourparameterchoicesaremainlyaccordingtothecharacteristicsofthedatacollectedfromtheED,whichisrepresentedbythepreemptivecase.Inordertoeliminatetheeectsthatmaybeduetoaspecichealthcarefacility,weusethesamesetofparametersforthenonpreemptivecasealso. Inourstudy,weconsideredthreeexperimentalfactorsincludingthefractionofclass1patients,f1t,theunitpatienttreatmentcostforeachpatientclass,(UP1t;UP2t),andthemaximumallowableaveragetimeinsystemforeachpatientclass,(1;2).EachparameteristestedatthreelevelsaslistedinTable 36 .Foreachexperimentalsetting,wegenerated25randominstancesofthepatientarrivalstreamsovertheplanninghorizon.WesolvedalltheinstancesforeachofthesettingsbyusingtheHCTSCPmodelwiththeAGEmethodandwiththeDBAmethodtocomparetheperformanceofthetwomethods. Table36. Parametersettings ParametersLevel1Level2Level3 Experiment1f1t(%)808590 Experiment2UP1t($/hour/patient)200400800UP2t($/hour/patient)50100200 Experiment31(hours)4.504.755.002(hours)3.504.004.50 63 PAGE 64 3.3 ,webuildanetworktorepresenttheHCTSCPproblem.Tocomputethepatientdelaycostforeacharc,weobtainaveragetimeinsystemforeachpatientclassapproximately(exactly)usingtheDBA(AGE)method.WenotethatwedonotusetheGEmethodinthisexperiment,sincetheAGEmethodisshowntobeconsiderablymoreecientinSection 3.5.1 .Afterweconstructthenetwork,wendtheshortestpathusingDijkstra'salgorithm[ 2 ].Theresultsshowthatthenetworkforthe12periodHCTSCPproblemwithsixandfourfeasiblecongurationsfortype1andtype2teams,respectively,contains290nodesand6,384arcs.Theaveragetimerequiredtoobtaintheoptimalcapacityplanis11.0secondsforpreemptivecaseand549.8secondsfornonpreemptivecaseifweusetheAGEmethodinbuildingthenetworkforaninstanceoftheHCTSCPproblem.Incontrast,theDBAmethodrequires0.03seconds,onaverage,forboththepreemptiveandnonpreemptivecases.Therefore,usingtheDBAmethodinbuildingthenetworkforaninstanceoftheHCTSCPproblemsisconsiderablymoreecientthanusingtheAGEmethod. InordertomeasuretheaccuracyoftheDBAmethod,wecomparetheteamcapacityplansgeneratedbythetwoapproachesbasedonasimilarityindex.Specically,wecountthenumberofperiodswhereteamcongurationschosenarethesameinbothapproachesandthendividethiscountingby12foreachcareteamtypetodeterminethevalueofthesimilarityindex.TheresultsofourcomparisonforeachexperimentalfactorsaresummarizedinTables 37 38 ,and 39 .Cellsrangefrom0percent(absolutelydierent)to100percent(perfectlysimilar).Forinstance,ifacellhasavalueof75percent,thenamong12periods,theHCTSCPmodelwithDBAgivesthesameresultsin8ofthemastheHSCTSCPmodelwithGE. InTable 37 ,weobservethattheHCTSCPmodelwithDBAworksverywellinthepreemptivecase.However,inthenonpreemptivecase,asthefractionofclass1patientsincreases,theperformanceoftheHCTSCPmodelwithDBAdeterioratesasitoverestimatestherequiredservicecapacityoftype2teambychoosingtheteam 64 PAGE 65 Impactofthefractionofclass1patientsonthesimilarityindex PreemptiveNonpreemptive 80%100.0%99.0%100.0%88.0%85%100.0%98.3%100.0%84.7%90%100.0%98.0%100.0%82.7% Table38. Impactofunitpatientdelaycostonthesimilarityindex PreemptiveNonpreemptive 200100.0%97.0%100.0%99.7%100.0%99.7%100.0%73.3%100.0%84.0%100.0%83.7%400100.0%97.0%100.0%99.7%100.0%99.7%100.0%74.3%100.0%84.7%100.0%84.3%800100.0%97.0%100.0%99.7%100.0%99.7%100.0%86.0%100.0%86.7%100.0%86.7% UnitofUP1tandUP2t:$/hourperpatient. Table39. Impactofmaximumallowableaveragetimeinsystemonthesimilarityindex PreemptiveNonpreemptive 6100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%7100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%8100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0%100.0% Unitof1and2:hourperpatient. PAGE 66 Table 38 showsthattheunitpatientdelaycostdoesnotimpacttheaccuracyoftheHCTSCPmodelinthepreemptivecase.Butinthenonpreemptivecase,itsaccuracyindeterminingtheteamcongurationoftype2teamgoesdown,asunitclass1patientdelaycostorunitclass2patientdelaycostdecreases,andittendstooverestimatethetherequiredservicecapacityofthetype2teamby1level.Table 39 showsthatthemaximumallowableaveragetimeinsystemdoesnotmateriallyimpacttheaccuracyoftheHCTSCPmodelineitherthepreemptivecaseorthenonpreemptiveone.Insummary,theHCTSCPmodelwithDBAisecientandaccurateinsolvingthecapacityplanningproblems,particularlyinthepreemptivecase,e.g.,theEDapplication.Forthenonpreemptivecase,e.g.,theOCapplication,itsaccuracyintype2careteamrequirementsisnotasprecise. Inourwork,werstconsideredthecasewheretheclass1patientscanpreemptclass2patientsservedintype1E/Trooms,whichistypicalinemergencyroomsinhospitals. 66 PAGE 67 Wenotethatinourqueueinganalysis,weassumedexponentiallydistributedinterarrivalandservicetimestopreserveanalyticaltractability.Extendingourworktoconsidergeneralarrivalandserviceprocessesisapotentialareaforfutureresearch.Moreover,inourstudyweanalyzedsystemswherepatientsarecategorizedintotwoclasses.Althoughthisclassicationiswidelyusedinthehealthcareindustry,someclinicsandhospitalsfurtherclassifyeachoftheclassesintotwoormoresubclasses[ 100 ].Thepresenceofmultiplesubclassesineachpatientclassmagniestheproblemsizerapidlyandcannotbesolvedthroughgeneralnumericalmethods(e.g.,theGEortheAGEmethod).Therefore,developingapproximationproceduresforsuchsettingswouldbeoftheoreticalandpracticalinterest.Inourwork,wefocusedonthelongtermcapacityplanningofthehealthcareteams,treatingtheE/Troomsastheserversandassumingthattheserviceratesoftheroomsfordierentpatientclassesaregiven,androomallocationisxedovertheplanninghorizon.Formoredetailedanalysis,thesetoftasksrequiredforthediagnosis,monitoring,andtreatmentofapatientcanbemodeledusingaqueueing 67 PAGE 68 68 PAGE 69 Ineectiveallocationofexistingbedcapacityamongdierentmedicalserviceunitscanleadtoservicequalityproblemsforthepatientsalongwithoperationaland/ornancialinecienciesforthehospitals.AnarrivingpatientmaybedeclinedorplacedonholdbyanMCU,ifthereisnobedavailabletoaccommodatethepatientintheunit.Inthiscase,thepatientmaybesubjecttosamehealthrisksduetothenecessitytondanalternativehealthcareproviderorwaituntilabedbecomesavailable.Fromthehospital'sperspective,thepotentialrevenueiseitherlostordeferred,whichmayleadtosomenancialineciencies.Similarly,anarrivingpatientmaybeacceptedfortreatmentbutcanbeaccommodatedinanotherMCU(e.g.,anarrivingobstetricspatientcanbeboardedinneurosurgery).Inthiscase,thepatientmaybesubjecttosameunnecessaryhealthrisksduetothenecessitytobeboardedtogetherwithpatientswithmoredierent,possiblymoreserious,healthconditions.Fromthehospital'sperspective,thepotentialrevenueiscollectedbuttheoperationalcostsmayincrease,asthepatientmaybeboardedinanMCUwheretheserviceresourcesaremoreexpensive(e.g.,neurosurgerynursesmayhaveadditionalqualicationsthanobstetricsnursesandthemedicalequipmentinaneurosurgerydepartmentaretypicallymoreexpensivethantheonesinobstetrics). Toeectivelyutilizeexistingbedcapacity,hospitaladministrationcouldchoosefromamonganumberofalternativeplanningstrategiestotondanallocationofexisting 69 PAGE 70 Inthiswork,wefocusontherststrategyanddevelopamathematicalprogrammingformulationtoaddressthisproblem.Wealsodevelopeectivesolutionapproachestoobtainhighqualitysolutionsparticularlyforlargesized,realistictestinstances. Theremainderofthischapterisorganizedasfollows.InSection 4.2 ,wereviewtherelatedliterature.Section 4.3 presentsmathematicalprogrammingformulationforHBA.WedevelopthreeheuristicsolutionapproachesinSection 4.4 .InSection 4.5 ,wepresentresultsfromourcomputationalstudythatevaluatesthecomputationalperformanceoftheapproachesdevelopedinSection 4.4 .Section 4.6 includesadiscussionoftheresultsandsuggestsfutureresearchdirections. 46 ].Totakethestochasticnatureofhealthcaresystemsintoaccount,researchersutilizequeuingandsimulationmodelsindeterminetheappropriatebedcapacityconguration. Theapplicationofqueueingtheoryallowsfortheevaluationoftheexpected(longrun)performancemeasureofasystembysolvingtheassociatedsetofowbalanceequations.MackayandLee[ 80 ]evaluatethechoiceofmodelsforforecastingbedcapacityandsuggesttousecompartmentalowmodel,whichmodelspatientowthrougha 70 PAGE 71 Gorunescuetal.[ 43 ]modeladepartmentofgeriatricmedicineasanM=M=c=Kqueueingsystemtoinvestigatetheinterrelationshipsbetweenadmissionrates,lengthofstay,numberofallocatedbeds,andprobabilitythatanarrivingpatientisdeniedadmission.KaoandTung[ 63 ]presentanapproachforallocatingbedstocareunitsinahospitaltominimizetheexpectedpatientoverow,i.e.,therateatwhichpatientsaredeniedadmissionduetoinavailabilityofbedcapacity.TheymodeleachserviceasanM=G=1queueingsystemandusenormalapproximationincomputingnumberofpatientoverows.Thebedallocationproblemissolvedintwostages.Therststagedistributesthemajorityofbedssuchthatnogrossimbalancesinbedutilizationamongallcareunitsareobservedandaprespeciedfractionofpatientscanstayintheunitsdesignatedfortheunit.Thesecondstageusesmarginalanalysistoallocatetheremainingbedstominimizetheexpectedtotalpatientoverow. Discreteeventsimulationisusefulfortheanalysisofsystemswithcomplexbehavior,includinghealthcaresystems.Discreteeventsimulationhasbeenwidelyappliedinhealthcareservices[ 58 ]tostudytheinterrelationshipsbetweenadmissionrates,hospitaloccupancy,andseveraldierentpoliciesforallocatingbedstoMCUs.Harrisonetal.[ 52 ]constructasimulationmodelwherepatients'stayinhospitalsareclassiedintothreestages,whichrepresentdierentphasesofcareprovided.Theoutputobtainedfromthemodelmatchesthemeanandthevariabilityassociatedwithactualbedoccupancydata.Themodelisusedtoidentifydailyoccupancydistributions,studytradeosbetweenoverowandbedcapacitylevels,andinvestigatetheeectsofvariouschanges.AkkermanandKnip[ 6 ]useMarkovchainapproachtospecifythenumberofbedsneeded 71 PAGE 72 84 ]usediscreteeventsimulationtoinvestigatetheinterrelationshipsbetweenbedoccupancy,averagenumberofpatientdeferred,anddierentbedallocationandoperatingpolicies.Harper[ 51 ]developsasimulationmodelfortheplanningandmanagementofhospitalbeds,operatingtheaters,andworkforceneeds.Themodelcapturesthecomplexityofhealthcaresystemsbyincorporatingthevariabilityforeachpatientgroupsuchasmonthly,daily,andhourlydemandaswellasthedistributionsoflengthofstayandoperationtimes.Kimetal.[ 68 ]analyzeanintensivecareunitwith14bedsanddevelopasimulationmodeltoevaluatedierentbedreservationschemestoreducethenumberofcancelledsurgeries. Somepapersconsiderthehierarchicalrelationbetweencareunits.Forexample,afteramothertobedeliversherchildinthelaboranddeliveryunit,sheshouldbemovedtothepostpartumunitforrecovery.Whenthecapacitydownstreamisinsucient,patientsareforcedtostayatthecurrentcareunitswithtypicallymoreexpensiveequipmentblockingthecapacityattheseupstreamcareunits.Totaketheinteractionsamongcareunitsinahospitalintoaccount,CochranandBharti[ 30 ]rstapplyqueueingnetworkmethodology(withoutblocking)tondabalancedbedallocation,whichisobtainedthroughtrialanderrorwork.Then,theyusesimulationanalysistoestimatetheblockingbehaviorandpatientsojourntimes.Galv~aoetal.[ 39 40 ]applyathreelevelhierarchical,capacitatedmodeltodeterminethecapacityrequiredinperinatalhealthcarefacilities,whichiscategorizedintothreelevels:basicunits,maternityhomes,andneonatalclinicswhereintensivecareunitforbabiesisavailable. Inourwork,weutilizeadierentapproachbyintegratingresultsfromqueueingtheoryintoanoptimizationframework.Specically,wemodeleachMCUinahospitalasanM=M=c=cqueueingsystemtoestimatetheprobabilityofrejectionwhentherearecbedsintheunit.Wethendevelopanoptimizationmodeltoallocatetheaggregatebed 72 PAGE 73 minDXi=1jij 73 PAGE 74 4{1 )minimizesthetotaldeviationofthebedoccupancyforeachoftheMCUsfromtheoverallaveragebedoccupancyforthehospital.Constraint( 4{2 )limitsthetotalnumberofbedsallocatedamongtheMCUstothenumberoftotalbedsavailableinthehospital.Constraintset( 4{3 )representstheprobabilityofrejectingpatientsarrivingtoserviceiasafunctionofarrivalratei,servicerateiandbedcapacityxiforservicei.Constraintset( 4{4 )representsthebedoccupancyoftheserviceiasafunctionofeectivearrivalrate(1pi)i,servicerateiandbedcapacityxiforeachservicei.Constraint( 4{5 )speciestheoverallaveragebedoccupancyforthehospital.Constraintsets( 4{6 )and( 4{7 )imposethelowerandupperboundsonthebedcapacityandthebedoccupancyforeachservicei,respectively.Finally,constraint( 4{8 )ensuresthatthedecisionvariablesarenonnegativeintegers. HBAisadicultnonlinearbinaryintegerprogrammingproblemwithnonlinearconstraints.Notethatconstraintset( 4{3 )involvesthefactorialfunctiononxiandthesummationoftheterm(i=i)m Asaresult,wecanobtainanequivalentlinearbinaryintegerprogrammingproblemthatcanbestatedasfollows: minDXi=1uiliXj=1ijyij 74 PAGE 75 Wehavefoursetsofadditionalconstraints.Constraintset( 4{15 )ensuresthatonlyonesplittingvariabletakesthevalueofone.Constraintsets( 4{16 )and( 4{17 )computethedeviationofthebedoccupancyforeachserviceifromtheoverallaveragebedoccupancyforthehospital.Constraintset( 4{18 )ensuresthatthedecisionvariablestakebinaryvalues.Finally,constraintset( 4{19 )providesinformationofthediscreteoptionsofxij. 41 53 ].Inparticular,twomembersofthecurrentpopulationofsolutionsarechosenrandomlyandusedtogeneratenewospring 75 PAGE 76 41 depictsthepseudocodeoftheGA.Apreprocessingstepensuresthatatestinstanceisfeasiblebyverifyingthevalidityofthefollowinginequalities: where(li)representsthebedoccupancywithbedcapacityliofservicei. endGeneticAlgorithm. Figure41. Pseudocodeofthegeneticalgorithm GAstartsbygeneratingasetofdistinctfeasiblesolutionstoformaninitialpopulationofsolutions.Thesesolutionsaregeneratedthroughanoccupancydrivenapproach,andthenadjustedtofeasiblebyarandomrectiedapproach(seeFigure 42 ). TheoccupancydrivenapproachrsttreatseachMCUasanM=M=cqueueingsystem,andthenallocatesanumberofbedstotheMCUsuchthatthebedoccupancyiscloseto(seeFigure 43 ).Asitcanbeobservedfromconstraintset( 4{3 ),specifyingtheprobabilityofrejectingapatientfromanMCUthatismodeledasanM=M=c=cqueueingsystemrequirestheknowledgeofthenumberofbedsallocatedtothatparticularMCU.SinceourobjectiveistogeneratesomesolutionsfortheinitialpopulationofGA,rather 76 PAGE 77 Figure42. Pseudocodeofthepopulationgeneratingprocedure thanspendingtimetondthecorrespondingbedcapacityforanM=M=c=cqueueingsystem,wemodeltheMCUasanM=M=cqueueingsystemtoinitializethebedcapacity. Figure43. Pseudocodeofoccupancydrivenallocation Thesolutionsgeneratedfromtheoccupancydrivenapproachmaynotsatisfyconstraintsets( 4{2 ),( 4{6 )and( 4{7 ),i.e.,constraintsetsassociatedwiththetotalnumberofbedsavailable,lowerandupperboundsandbedoccupancy(ofanM=M=c=cqueueingsystem).Weusearandomrectiedproceduretonetunetheinitialbedallocation.Figure 44 showsthepseudocodeoftherandomrectiedprocedure.WerstrevisethebedcapacityofeachMCUsuchthattheconstraints( 4{6 )and( 4{7 )aresatised.Then,whenthetotalnumberofbedsallocatedisgreater(less)thanB,anMCUischosenrandomlyanditsbedcapacityisdecreased(increased)byone,ifthisoperationdoesnotviolateconstraint( 4{6 ).ThestepofchoosinganMCUrandomlyandadjustingitscapacityaccordinglyisrepeateduntilthetotalnumberofbedsallocatedisequaltotheoverallbedcapacityavailable,B. 77 PAGE 78 Figure44. Pseudocodeofrandomrectiedprocedure Aftertheinitialpopulationisformed,GAstartstoproduceospring.Twotypesofgeneticoperatorsareconsideredinthiswork;oneissimplesinglepointcrossover,andtheotherismutation.Insimplesinglepointcrossover(seeFigure 45 ),werstrandomlyselecttwoparentsfromthepopulation,andthenswapbedcapacitybetweentherandomlyselecteddD=2eMCUs.Thatis,iftheMCUiisselected,thebedcapacityofMCUioftherstparent'schromosomeisswappedwiththebedcapacityofMCUiofthesecondparent'schromosome.Theresultingsolutionsaretheospringsolutionsproducedbytheselectedparentsolutions.Figure 46 showsanexamplewithfourMCUsand100beds,wherethesecondandthethirdMCUsarechosentoswaptheirbedcapacity.Notethatthecrossovermayproducedanospringwhichdoesnothaveachromosomespecifyingafeasiblesolution.Inthiscase,therandomrectiedapproachisemployedtoadjustthebedcapacityallocation. Thesecondgeneticoperator,mutation,isinvokedifthecurrentbestsolutionisnotimprovedthroughouttheevolutionofaprespeciednumberofconsecutivegenerations.Themutationemploystheoccupancydrivenapproachwiththeaverageoccupancy 78 PAGE 79 endCrossover. Figure45. Pseudocodeofcrossoverprocedure Figure46. Exampleofcrossover ofthecurrentbestsolutiontoproduceanospring.Again,thismutatedospringisadjustedthroughtherandomrectiedproceduretoensurefeasibility.Then,thevalueofNumGenNotImproveissettozero. Iftheospringproducedfromeithercrossoverormutationproceduredoesnotexistinpopulationandisbetterthanthecurrentworstsolutioninpopulation,thenthecurrent 79 PAGE 80 Figure47. Pseudocodeofthemutationprocedure worstsolutioninthepopulationisreplacedwiththeospringtoformanewgenerationwiththeotherexistingsolutions.Otherwise,i.e.,eithertheospringexistsinpopulationalreadyortheospringisnotbetterthanthecurrentworstsolutioninthepopulation,thisospringisignoredandthenextospringisproducedusingtheproceduredescribedabove.Notethatthecurrentbestsolutionisalsoupdatedifthenewospringhasabetterobjectivefunctionvaluethanthatofthecurrentbest. 36 ]thatiswidelyusedforcombinatorialoptimizationproblems.EachGRASPiterationconsistsoftwophases:constructionandlocalsearch.Figure 48 depictsthepseudocodeforGRASP.Theconstructionphasecreatesafeasiblesolution,whoseneighborhoodisexploredbythelocalsearchphasetondalocallyoptimalsolution.Thealgorithmstopsafteraprespeciednumberofiterationsisexecuted,whichisdenotedbyMaxNumIterationsinFigure 48 endGRASP. Figure48. PseudocodeofGRASP 80 PAGE 81 49 ,wheretheejinline 22 representsazerovectorexceptthejthelementequalsone.Lines 15 through 20 inFigure 49 buildarestrictedcandidatelist(RCL),whichrecordsthesetofMCUsforwhichaddingonemorebedtotheMCUdoesnotviolatethefeasibilityandhasthepotentialtoimprovetheobjectivefunctionvalueatanacceptablelevel.Thethresholdofincrementontheobjectivefunctionvalueiscontrolledbytheparameter2f0;1gonline 17 inFigure 49 .AnMCUisincludedinRCL,ifaddingonemorebedtotheunitisfeasible,andtheincrementontheobjectivefunctionvalueisnotgreaterthanmin+(maxmin),whereminandmaxrepresenttheminimumandmaximumincrementsonobjectivefunctionvalueafterincorporatingonemorebedtocurrentsolution,respectively.Notethatthelowertheis,thegreediertheprocedureis.AbedisaddedtoanMCUselectedrandomlyfromRCL,untilthetotalnumberofbedsallocatedequalsB. Duringthelocalsearchphase,theneighborhoodofthefeasiblesolutioncreatedintheconstructionphaseisfullyinvestigatedtondthelocaloptimum.AneighborsolutionisproducedbyswappingonebedfromoneMCUtoanother,ifthisswapdoesnotdestroythefeasibilityofthesolution.ThepseudocodeofthelocalsearchprocedureisillustratedinFigure 410 .Here,weadoptthebestimprovingstrategythatisweevaluateallfeasibleneighborsandchoosetheneighborthatimprovestheobjectivefunctionvaluethemost. 411 .OneachiterationoftheHA,asetofelitesolutionsisgenerated,incontrasttoasingleinitialsolutionusedbyGRASP.Then,anospringis 81 PAGE 82 Figure49. Pseudocodeofgreedyrandomizedconstructionprocedure producedusingthecrossoverprocedureofGAandimprovedbythelocalsearchprocedureofGRASP.Theprevioustwostepsarerepeated,untilaprespeciednumberofiterationsarecompleted.ThecharacteristicoftheHAisthatoneachiterationthethresholdforthegreedyrandomizedconstructionphaseisupdatedbyadecreasingfunctionp(n),i.e.,decreasesasthenumberofiterations,n,increases.Inthisway,theHAcantestmorethanonesettingofandreducetheprobabilityofconvergingtoalocaloptimumsolutionprematurely. Toformanelitesetofsolutions,werstusetheGreedyRandomizedConstructionprocedureofGRASPtogenerateNumGRASPdistinctsolutions,andthenchoosethebestESizeofthem,whereNumGRASP>ESize.ThepseudocodeispresentedinFigure 412 82 PAGE 83 Figure410. Pseudocodeoflocalsearchprocedure endHybrid. Figure411. PseudocodeofHA 83 PAGE 84 Figure412. Pseudocodeofelitesetgenerationprocedure wherefAdenotestheobjectivefunctionvalueobtainedbyusingapproachA,whereA2fGA;GRASP;HAg. Inourstudy,weconsiderthreesettingseachwithdierentinstancesizestoevaluatetheimpactofthesizeoftheinstanceontheperformanceoftheproposedsolutionapproaches.Theproblemsizeisvariedwiththenumberoftotalbedsavailableinthehospital,B,andnumberofMCUsinthehospital,D.ForthetotalnumberofbedsandthenumberMCUsavailableinthehospital,weconsiderthreelevelsthatcorrespondtosmall,medium,andlargesizedhospitals.Inparticular,weconsiderhospitalswith750,1,000or1,250bedsand30,40or50MCUs.Foreachsetting,wegenerate30randominstances.EachrandominstanceisobtainedbygeneratingparametersthatcorrespondtopatientarrivalandserviceratesalongwithlowerandupperboundsonthenumberofbedsavailableineachMCU.Specically,eachparameterisobtainedbyusingtheformula:(meanvalue)u,whereuisdrawnfromthedistributionU[0:4;1:6].Themeanvaluesoftherandomparametersarelistedasfollows: 1.Meanarrivalrate(i):10persons/perunittimeforeachMCUi; 2.Meanservicerate(i):0.5persons/perunittimeforeachMCUi; 3.Meanlowerbound(li):10bedsforeachMCUi;and 84 PAGE 85 Last,theminimumbedoccupancy,,is70%forallMCUallproblems.TheexperimentisimplementedonaworkstationwithtwoPentium43.2GHzprocessorand6GBofmemory. Toobtainanearoptimalsolutionasacomparisonbasis,weuseCPLEXtoobtaintheoptimalsolutionsforthetestinstances.Foreachtestinstance,CPLEXisstoppediftherelativestoppingtoleranceof0.01%issatised,orCPUtimeof3,600secondsisused.Table 41 reportstheresultsobtainedbyCPLEX.Asweexpect,thenumberofinstancesforwhichCPLEXcandetermineandverifytheoptimalsolutionwithinonehourofCPUtimedecreasesastheproblemsizeincreases.Moreover,fortheinstanceswithsmallproblemsize,CPLEXtakesmorethan1,800secondsonaveragetosolveoneinstance. 42 reportstheresultsofGA,wheretheinformationisclassiedintotwolayers.Therstlayerdistinguishestheinstanceswithdierentsizes,andthesecondspeciestheparametersettingofGA,includingnumberofgenerationsandpopulationsize.Notethatiftherelativeerrorofaninstanceislessthanzero,itisreplacedbyzerowhencomputingtheaveragerelativeerrorofthecorrespondinginstance. Ingeneral,GAperformsquitewellintermsofaveragerelativeerrorandCPUtime.Foranyofthe90instances,GAspendslessthan2secondstondasolutionandtheaveragerelativeerrorislessthan2%.Furthermore,formorethanonethirdofthetestinstances,GAndssolutionsbetterthanCPLEX.Forexample,forthefourthlargesizedinstance,GAndsasolutionwithanobjectivefunctionvaluewhichisabout5.6%lowerthanthatofthesolutionfoundbyCPLEX. 85 PAGE 86 NearoptimalsolutionsobtainedbyusingCPLEX InstanceObjectiveTime(sec.)ObjectiveTime(sec.)ObjectiveTime(sec.) 10.1964370.26913180.5298>3,60020.613240.4122>3,6001.0745>3,60030.2574>3,6000.6417>3,6000.5083>3,60040.28691,9720.3864>3,6000.7238>3,60050.4828>3,6000.4141>3,6001.0353>3,60060.4588210.7636>3,6000.2228>3,60070.28171,3690.3180>3,6000.5841>3,60080.24847090.5691>3,6000.4073>3,60090.13512,1110.4530>3,6000.6679>3,600100.45581100.3737>3,6001.1034>3,600110.34471,2770.3347>3,6000.5721>3,600120.4320470.4972>3,6000.6960>3,600130.2872>3,6000.6381>3,6000.4622>3,600140.458940.7216>3,6000.7431>3,600150.5883>3,6000.6496>3,6000.4568>3,600160.28483750.8077>3,6000.9320>3,600170.2751>3,6000.5183>3,6000.5609>3,600180.1942>3,6000.20225430.5702>3,600190.3408>3,6000.6919290.7131>3,600200.260080.4640>3,6000.6928>3,600210.3262>3,6000.6155>3,6001.0242>3,600220.4457>3,6000.52051,7270.6448>3,600230.181380.4363>3,6000.4886>3,600240.3991>3,6000.1789>3,6000.7533>3,600250.6148>3,6000.3602>3,6000.7442>3,600260.3832>3,6000.7231>3,6000.4640>3,600270.8989120.7193>3,6000.5765>3,600280.33927300.5233>3,6000.7700>3,600290.13771,7700.6959>3,6000.4138>3,600300.4448130.48335771.0881>3,600 No.ofinstancessolvedbyCPLEX1850 86 PAGE 87 SolutionsobtainedbyGA MediumLargesize No.of6000 60006000 800080008000 100001000010000generationsPopulation30 6090 4080120 50100150size Err.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeInstance (%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.) 1 0.000.560.000.570.000.580.001.000.001.000.001.026.261.566.261.606.261.682 9.280.589.280.609.280.623.861.013.861.043.861.062.101.602.101.592.101.673 0.000.570.000.580.000.600.001.030.001.040.001.060.001.560.001.590.001.674 0.000.560.000.570.000.571.741.001.741.031.741.045.581.525.581.565.581.645 0.000.600.000.610.000.610.000.980.000.990.001.010.691.560.691.580.691.656 0.000.550.000.560.000.573.091.043.291.073.311.090.001.580.001.620.001.707 0.000.570.000.580.000.590.000.990.001.010.001.020.001.590.001.640.061.738 0.000.550.000.560.000.570.001.031.411.060.001.090.001.580.001.620.001.709 0.000.570.000.570.000.580.001.030.001.030.001.040.881.601.281.691.871.7710 0.000.560.000.580.000.580.440.990.441.020.441.047.061.607.131.668.201.7411 0.000.570.000.580.000.590.001.020.001.040.001.060.001.560.001.610.001.6812 0.000.570.000.590.000.620.001.000.061.020.061.035.781.585.781.605.781.6613 0.000.560.000.560.000.570.900.990.901.000.901.020.001.570.001.600.001.6814 3.450.553.450.563.450.579.450.999.451.009.451.020.111.580.111.620.111.7115 0.000.570.000.580.000.580.001.010.001.020.001.040.001.540.001.550.001.6316 0.000.560.000.570.000.590.001.000.001.030.001.041.821.562.001.592.351.6717 0.000.570.000.570.000.580.001.010.001.030.001.051.091.601.091.641.091.7318 0.000.580.000.610.000.620.001.040.001.050.001.080.591.600.591.660.411.7619 0.000.570.000.570.000.583.431.042.491.084.681.110.661.580.661.610.671.7020 0.000.600.000.620.000.640.001.019.341.029.341.050.911.581.041.614.341.6921 0.000.570.000.570.000.590.001.020.001.050.001.070.271.550.271.590.001.6822 2.060.560.000.580.000.590.000.970.000.980.001.000.321.600.321.640.321.7323 0.010.560.010.560.010.560.001.030.001.060.001.091.011.611.081.661.101.7724 4.830.554.830.564.830.580.001.010.001.010.001.033.401.523.401.543.401.6225 2.820.573.480.583.550.590.001.010.001.040.001.060.001.530.001.551.101.6226 0.000.570.000.580.000.597.201.037.201.057.211.082.241.552.241.582.241.6527 0.000.560.000.570.000.582.061.002.061.022.061.050.621.561.001.601.961.6928 0.000.580.000.590.000.600.001.020.001.050.001.083.761.563.691.582.311.6729 0.000.580.000.600.000.625.401.005.401.025.401.040.001.530.001.570.001.6330 4.950.595.030.605.220.620.001.020.001.040.001.072.451.603.041.642.501.73 Min. 0.000.550.000.560.000.562.060.972.060.982.061.005.581.525.581.545.581.62Avg. 0.910.570.870.580.880.591.181.011.521.031.551.051.111.571.191.611.401.69Max. 9.280.609.280.629.280.649.451.049.451.089.451.117.061.617.131.698.201.77 No.of13 1313 111011 121010instancesGAndsbettersoln.thanCPLEX PAGE 88 43 displaystheresultsofGRASP.AsisobvioustoseethatGRASPtakesCPUtimeslongerthanGAtoobtainsolutions.GRASPspends3.1secondsinaveragetoobtainasolutionoftheproblemwithsmallsize,incontrasttoGA's1.6secondswithrespecttotheproblemwithlargesize.Ingeneral,theaccuracyofGRASPandGAdoesnotappearsignicantlydierent,ifwecomparethebestresultsfromeachofthem.Forexample,intheproblemwithmediumsize,therelativeerrorsofGRASP(with=0.25)are1.37%and8.24%inaverageandmaximum,respectively,andGA(withpopulationsize=40)are1.18%and9.45%,respectively.Furthermore,among30instanceGRASP(with=0.25)ndsresultsbetterthanCPLEXfor9instances,comparedto11instancesofGA(withpopulationsize=40). Table 43 presentsthetendencythattheloweris,thelowertheaveragerelativeerroris.However,alowvalueofhasthedrawbackthatitmayleadthesolutiontoalocaloptimalsolutionandresultinalargeerror.Forexample,theinstance20inproblemwithmediumsizehasrelativeerror8.24%whenis0.25.Incontrast,theinstancehaslessrelativeerror,3.77%,whenisincreasedto0.35. InGAandGRASP,wedoobservethatsomeinstanceshaverelativeerrorgreaterthan8%,suchastheinstance14oftheproblemwithmediumsizeinTable 42 ,andtheinstance20oftheproblemwithmediumsizeinTable 43 .Toreducetheerrorsofthoseinstances,wecombinesthesetwoalgorithmstodevelopahybridization. 88 PAGE 89 SolutionsobtainedbyGRASP Medium Largesize No.of600 600600 800800800100010001000starts0.25 0.300.35 0.250.300.35 0.250.300.35 Err.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeInstance (%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.) 1 0.003.000.002.960.002.940.008.900.008.840.009.396.3322.786.7622.518.5624.192 0.983.242.813.182.813.154.369.104.578.957.919.212.1023.722.1023.542.1024.003 0.002.920.002.900.002.880.009.830.009.750.0910.390.0023.310.0622.981.7324.554 0.003.220.003.210.003.202.769.535.279.525.2710.035.5823.505.5823.345.5824.845 0.003.200.003.141.083.130.009.210.009.130.009.360.4724.360.9524.211.2725.026 0.783.270.783.230.783.222.889.653.319.5411.1910.020.0022.300.0022.183.0623.717 0.003.090.003.050.003.050.009.840.009.760.0010.340.4922.440.9422.102.5223.938 0.003.130.003.080.003.060.009.350.009.200.149.420.0024.020.1723.851.4825.319 0.003.540.003.520.003.511.149.631.489.532.3210.192.6822.473.0221.985.0422.5910 0.002.930.002.930.002.930.449.720.449.640.4410.116.0822.778.6522.4419.5923.9511 0.003.030.002.990.002.970.009.920.009.850.0010.360.0024.210.0023.900.0025.4312 0.003.090.093.061.113.030.069.580.069.390.069.785.7823.755.7823.525.8723.9313 0.003.170.003.130.003.130.909.710.909.660.9010.150.0023.870.0023.660.0024.8014 0.143.390.143.350.143.327.529.607.889.589.0010.250.5023.254.8123.0317.5824.2115 0.003.120.003.100.003.110.189.970.189.800.5710.120.0023.550.0023.440.0024.8716 0.003.250.003.212.083.200.429.450.099.322.389.633.2023.913.9023.663.8625.1417 0.003.450.003.410.003.360.009.840.009.750.7110.030.5123.972.1423.854.8325.0618 0.003.410.003.373.883.330.009.170.009.010.009.550.0923.441.3023.194.8824.7119 0.002.990.002.950.002.941.738.962.668.864.929.170.6621.970.7521.781.4923.2120 0.003.220.003.160.003.138.249.366.019.193.779.504.4323.144.4622.868.0823.2021 0.003.330.003.330.003.290.009.520.449.3718.199.911.4823.211.4222.927.3724.8322 0.503.100.403.060.003.041.279.541.279.461.149.741.0821.520.9721.085.5422.3323 0.013.170.013.170.013.140.039.390.399.272.239.521.2823.232.5322.9511.6024.4224 4.833.194.833.144.833.130.0010.010.009.870.0010.513.2724.743.2824.543.3026.1625 5.513.206.173.195.833.170.009.680.009.530.009.816.2622.882.7722.536.5424.1326 0.003.220.003.180.003.174.569.494.599.409.399.670.5023.290.5023.160.5024.5727 7.043.317.043.267.043.252.069.532.069.422.069.882.6323.571.2423.281.3724.5328 0.003.134.003.0611.953.030.059.110.669.002.159.298.3823.498.0623.088.8424.0529 0.003.010.002.980.002.935.689.775.529.706.2310.290.0024.000.0023.920.0025.3630 3.742.951.112.933.982.910.009.500.259.346.919.881.9822.792.0122.255.6924.07 Min. 7.042.927.042.907.042.882.068.902.068.842.069.175.5821.525.5821.085.5822.33Avg. 0.523.180.653.141.253.121.379.531.499.423.169.851.8723.322.1523.064.6724.37Max. 5.513.546.173.5211.953.518.2410.017.889.8718.1910.518.3824.748.6524.5419.5926.16 No.of15 1312 986 875instancesGRASPndsbettersoln.thanCPLEX PAGE 90 Constant:p1(n)=0:3;Linear:p2(n)=1:00:7n N;andNonlinear:p3(n)=1:00:7r N Toprovideabenchmark,weincludetheconstantfunctionp1(n)inourexperiment,whereissetto0.3oneachiterationnforn=1,2,...,bD=10c.Usingfunctionp2(n),welinearlydecreaseto0.3asthenumberofiterationsincrease.Usingfunctionp3(n),wedecreaseto0.3inanonlinearfashionovertheiterations.Notethatvaluesgeneratedbyfunctionp3(n)isnolargerthanthosegeneratedbyfunctionp2(n)foreachnasshowninFigure 413 Figure413. Theupdatingfunctionsof 44 .Ineachtestinstance,therstcolumnreportstherelativeerrorsandCPUtimesobtainedbyusingtheupdatingfunctionp1(n),andsoon.Forthemediumsizedtestinstances,theaverage(maximum)relativeerrordecreasesfrom1.28%(9.34%)to0.35%(4.56%),whenp2(n)isusedinsteadofp1(n).Thefunctionp3(n),whichhassmallerthanp2(n)ateachiterationn,doesnotappeartomakeHAperformbetterthantheHAwithp2(n).Forexample,theaveragerelativeerrorforlargesizedinstancesincreasesfrom0.35%to0.44%,ifthep2(n)isreplacedbyp3(n).Thisisdierent 90 PAGE 91 SolutionsobtainedbyHA Medium Largesize No.of3 33 444555iterationsNo.of6000 6000 600080008000 80001000010000 10000generationsp1(n) 3030 404040 505050 Err.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeErr.TimeInstance (%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.) 1 0.007.940.007.930.007.920.0027.650.0027.660.0027.356.2671.110.0071.860.0072.512 2.817.650.007.710.007.663.8625.310.0025.800.0026.182.1062.192.1064.492.1063.263 0.007.180.007.700.007.440.0024.220.0024.560.0024.790.0077.700.0075.530.0077.464 0.008.070.008.170.007.951.7425.751.7425.521.7425.845.5874.415.5873.715.5874.295 0.007.290.007.500.007.350.0025.760.0023.710.0025.260.6966.540.6965.090.6965.696 0.588.000.008.140.008.052.5525.060.0025.680.0025.810.0077.340.0077.190.0077.767 0.007.390.007.740.007.330.0027.990.0028.060.0028.290.0070.420.0073.270.0072.768 0.007.940.007.070.007.910.0024.650.0024.630.0024.740.0072.010.0073.610.0074.519 0.008.180.008.360.008.130.0026.050.0026.450.0026.412.6870.812.6869.480.0070.4510 0.007.040.007.270.007.080.4425.870.4425.310.4425.702.6067.491.3567.591.5267.8311 0.007.800.007.930.007.750.0026.490.0026.420.0026.680.0075.010.0074.540.0075.3012 0.007.710.007.900.007.820.0626.630.0026.520.0626.805.7876.061.5775.611.5777.0113 0.008.220.008.260.008.220.9026.910.9026.800.9026.810.0072.490.0071.260.0072.3014 0.147.460.167.471.927.358.6525.372.8325.663.7625.740.1169.440.1170.550.1170.0615 0.007.330.007.710.007.400.0027.190.0027.200.0027.530.0078.020.0077.320.0078.3016 0.007.600.007.770.007.630.0023.770.0023.370.6824.461.6267.230.0067.170.0068.1217 0.008.140.008.400.008.150.0025.360.0025.250.0025.530.0069.200.7870.690.7871.0118 0.007.880.008.300.008.070.0027.850.0027.920.0028.140.0072.930.0073.510.0073.4119 0.007.460.007.880.007.511.4625.200.0025.070.0025.090.6670.790.0070.190.6671.4620 0.007.450.007.760.007.549.3426.680.0027.300.0027.424.3472.950.0074.364.3474.2821 0.007.400.007.680.007.470.0025.820.0025.030.0025.740.0066.710.2769.060.1468.9522 0.007.530.007.820.007.561.0526.080.0025.700.1426.350.0067.370.0068.340.0067.7223 0.016.190.016.380.016.170.0025.550.0026.180.0026.351.0169.361.0173.450.0072.9424 4.837.574.837.954.837.550.0028.010.0028.180.0028.523.4072.413.4071.363.4073.0825 2.827.802.828.072.827.830.0026.610.0026.600.0026.840.0071.320.0070.610.0071.8026 0.007.790.008.220.007.904.0325.864.5625.494.2325.742.2478.862.2477.072.2478.6227 6.057.375.917.326.707.082.0625.522.0625.512.0625.781.0675.351.0675.181.0675.1328 0.007.470.007.810.007.620.0023.740.0024.200.0024.463.7670.793.7670.873.7671.5329 0.007.920.008.230.007.975.4026.840.0027.172.0627.260.0078.760.0078.240.0079.1230 3.897.330.007.690.007.620.0024.800.0025.400.0025.350.0069.440.0068.750.0069.22 Min. 6.056.195.916.386.706.172.0623.742.0623.372.0624.465.5862.195.5864.495.5863.26Avg. 0.487.600.267.800.327.631.2825.950.3525.950.4426.230.9771.820.2972.000.3872.53Max. 4.838.224.838.404.838.229.3428.014.5628.184.2328.526.2678.863.4078.244.3479.12 No.of15 1515 111314 151716instancesHBndsbettersoln.thanCPLEX PAGE 92 HAoutperformsbothGAandGRASPintermsofsolutionquality.Therelativeerrorislessthan0.5%inaverageand5%inmaximumforallproblemswhetherp2(n)orp3(n)isused.Furthermore,HA(withp2(n)orp3(n))ndsbettersolutionsthanCPLEXforabouthalfoftheinstances,comparedtoGAorGRASP'sonethird.Astothecomputationalrequirement,HAspendslessthan80secondstoobtainasolutionforthelargesizedinstances,whichislargerthaneitherGAorGRASP,butisstillatanacceptablelevelfromapracticalperspective.Fromthese,wecanconcludethatHAisaneectiveandecientsolutionapproachfortheHBAproblem. Inourwork,eachMCUismodeledasanM=M=c=cqueueingsystem.Futureresearchcanconsiderdierentqueueingsystemcongurationswithgeneralarrivalandservice 92 PAGE 93 93 PAGE 94 Inahealthcareservicefacility,whentheemergencyroom(ER)isfullorallintensivecarebedsareoccupied,hospitalssendoutdivertstatus.Whenahospitalisondivertstatus,incomingpatientsmightbesenttohospitalswhicharefartherawayorkeptatthehospitalswheretheycurrentlyarethatmaynotabletoprovideadequateservice.Toacriticalpatient,theconsequenceofdivertstatuscanbethedierencebetweenlifeanddeath. Thepurposeofthisworkistoconstructafacilitylocationmodel,whichsimultaneouslydeterminesthenumberoffacilitiesopenedandtheirrespectivelocationsaswellasthecapacitylevelsofthefacilitiessothattheprobabilitythatallserversinafacilityarebusydoesnotexceedapredeterminedlevel.Inotherwords,wewanttolocateERservicesonanetworkanddeterminetheirrespectivecapacitylevelssuchthattheprobabilityofdivertingpatientsisnotlargerthanaparticularthreshold. Theremainderofthischapterisorganizedasfollows.InSection 5.2 ,wereviewtherelatedliterature.Section 5.3 presentsmathematicalprogrammingformulationforERSFLCP.Section 5.4 detailstheLagrangianrelaxationalgorithmthatweproposefortheERSFLCPproblem.InSection 5.5 ,wepresentresultsfromourcomputationalstudythatevaluatesthecomputationalperformanceoftheLagrangianrelaxationalgorithm.Section 5.6 includesadiscussionoftheresultsandsuggestsfutureresearchdirections. 94 PAGE 95 Thereisarichbodyofliteraturethatdeveloprobustfacilitylocationmodelsunderuncertaintytohedgetherandomnessoncosts,demands,andtraveltimesutilizingtherobustorstochasticoptimizationapproach.Snyder[ 103 ]presentsadetailedreviewofonstochasticandrobustfacilitylocationmodels. Intheareaofhealthcareapplication,Beraldietal.[ 12 ]investigatetheproblemofcharacterizingtheoptimallocationsofemergencymedicalservicesitesandnumbersofemergencyvehiclesrequiredforeachsite.Theyconsidertheproblemformulationinastochasticoptimizationsetting,wheretheabilitytocovertherandomrequestsofemergencyserviceatdemandpointsisrestrictedbyasetofprobabilisticconstraints.Specically,theprobabilisticconstraintsensurethattheprobabilitythatthenumberofvehicleslocatedatafacilitycancovertherandomservicerequestisgreaterthanaprescribedprobabilityvalue.Somepapersconsiderthehierarchicalrelationbetweenfacilitiesinahealthcaresetting.Forexample,Koizumietal.[ 70 ]classifymentalhealthcaresystemintolevelsofextendedacutehospitals,residentialfacilitiesandsupportedhousing,andpatientsowthroughthelevelsaccordingtotheirhealthconditions.Becauseofthehierarchicalstructureofhealthcareservice,thecapacityrequirementsofunitsinahigherlevelofhierarchyusuallycorrelatewiththeunitsinalowerlevel.Galv~aoetal.[ 39 40 ]formulateahierarchicallocationallocationmodeltodeterminethecapacityrequired 95 PAGE 96 Anotherbranchofliteraturethatisrelatedtoourworkdevelopsfacilitylocationmodelswhichrequirebackup(multiple)coverageforeachdemandpoint.SnyderandDaskin[ 104 ]consideraPmedianbasedmodelthatminimizestotalcostsassociatedwithalocationallocationplanandtheexpectedfailurecost.Aseachdemandpointisassignedprimaryandbackupfacilities,theexpectedfailurecostisquantiedbytheadditionaltransportationcostincurredtocoverthedemandbythebackupfacility.Theproblemisformulatedasa01integerprogrammingformulationandsolvedbytheLagrangianrelaxationalgorithm.Jiaetal.[ 56 ]presentadetailedreviewoftraditionalfacilitylocationmodelsandproposeageneralfacilitylocationmodelsuitedforlargescaleemergencies.Intheirmodel,ademandpointisconsideredtobecoveredifaprespeciednumberoffacilitiesareassigned. Toaddresstheissueofservicequality,somepapersincorporatequeuingsystemsintofacilitylocationmodelstoconsidertherandomnessonavailabilityofserversandfocusonreducingthedemandlostduetotheshortageofcapacityorsystemcongestion.MarianovandReVelle[ 81 ]formulateamaximalavailabilitylocationmodel,whichusesaM=M=c=cqueueingsystemtomodeltheserveravailabilityofademandpoint.Themodelwantstolocateasetofambulancessuchthatthedemandcoveredismaximized,whereademandpointisconsideredtobecovered,iftheprobabilitythatthereisanambulancenearbyandavailableisgreaterthanathreshold.Theyshowhowtotransformthenonlinearqueueingexpressiontoanequivalentlinearone,andsolvetheproblembyusingacommercialsolver(LINDO).Bermanetal.[ 13 ]modeleachfacilityasanM=M=1=aqueueingsystem,whereaisthemaximumnumberofcustomersallowedinthefacility,andconsiderafacilitylocationproblemwiththeupperboundsontheamountofdemandlostduetoinsucient 96 PAGE 97 14 ]alsoinvestigatethefacilitylocationproblemundertheobjectiveofmaximizingcaptureddemand.They,again,modeleachfacilityasanM=M=1=aqueueingsystemandassumethatacustomerislostiftheclosestfacilityandallotherfacilitiesthathe/orshecanreacharefull.Bothpapers([ 13 14 ])obtainthesolutions,i.e.,thelocationofthefacilities,throughheuristicapproaches. Wenotethatthecapacitiesoffacilitiestobeopenedhasanimpactonboththetotalnumberandthelocationsoffacilities,particularlywhenthecongestionassociatedwiththepotentialfacilitiesistakenintoaccount.Inwhatfollows,wemodeleachfacilityasanM=M=c=cqueueingsystem,wherecisnumberofserversinthefacility,whichdesignatesthecapacityofthefacility.Thegoalofourmodelistoidentifylocationsoffacilitiesandspecifythecapacitylevelsofthefacilitiessimultaneously. Letfjbethexedcostofopeningafacilityatnodej2N,andcjkbetheoperatingcostofthefacilityjwithcapacitylevelk2K,whereK=f1;2;:::;KgisthesetofcapacitylevelsofanopenedER.Inthisproblem,thecapacityismeasuredasnumberofbedsinanER.Thatis,ifanERistobeopenedatcapacitylevelk2K,thentherearemkbedsintheER.WeassumethatthelengthsofstayofapatientintheERatnodejareexponentiallydistributedwithratej,andeachERismodeledasanM=M=c=c PAGE 98 Wehavefoursetsofdecisionvariables.Therstsetispatientallocationvariablexij,whichtakesthevalueofoneifnodeiisservedbytheERservicefacilityatlocationj,zerootherwise.ThesecondsetisERlocationandcapacityallocationvariableyjk,whichtakesthevalueofoneifanERservicefacilityisplacedatnodejandoperatedatcapacitylevelk,zerootherwise.Thelasttwosetsarejandzj,whicharedeterminedoncethevaluesofxijandyjkareassigned. TheERSFLCPproblemcanbeformulatedasanonlinearintegerprogrammingformulationasfollows: minNXi=1NXj=1tnidijxij+NXj=1KXk=1(fj+cjk)yjk 98 PAGE 99 where(j;j;zj)istheprobabilityofdivertingpatientsfromanERservicefacilityatlocationj.Theformulasfor(j;j;zj)ofanM=M=c=cqueueingsystemcanbefoundinanyqueueingbook(e.g.,GrossandHarris,1998).Specically,(j;j;zj)canberepresentedasafunctionofarrivalratej,serviceratej,andbedcapacityzj,thatis(j;j;zj)=(j=j)zj Theobjectivefunction( 5{1 )minimizesthevalueoftimeofERpatientsandthecostsofopeningandoperatingERservicefacilities.Constraint( 5{2 )stipulatesthateachdemandnodemustbecoveredbyonefacility.Constraint( 5{3 )restrictsthatdemandnodescanonlybeassignedtoopenedfacilities.Constraint( 5{4 )imposestheupperboundonthenumberoffacilitiesopened.Constraint( 5{5 )statesthatanopenedfacilitymustbeassociatedwithasinglecapacitylevel.Constraints( 5{6 )and( 5{7 )obtainthearrivalratesandcapacitylevelsforallthefacilities.Constraint( 5{8 )imposestheupperboundontheprobabilityofallbedsbeingoccupiedforanopenedfacility.Finally,constraints( 5{9 )and( 5{10 )ensurethatdecisionvariablesarebinaryandnonnegativeintegers,respectively. Notethatthefactorialtermsinequation( 5{11 )maketheERSFLCPproblemintractable.Toovercomethisproblem,wereplacetheconstraint( 5{8 )byconstraint( 5{12 ), wherejkisthelargestvaluewhichsatisesinequality( 5{13 )andequation( 5{14 ).(jk;j;mk)=(jk=j)mk 99 PAGE 100 Asaresult,wetransformthepreviousnonlinearintegerprogrammingformulationtoalinearbinaryintegerprogrammingformulationasfollows: minNXi=1NXj=1tnidijxij+NXj=1KXk=1(fj+cjk)yjk 51 presentsthegeneralprocedureweusetoobtainasolutiontotheERSFLCPproblemusingaLagrangianrelaxationapproach.InFigure 51 ,nistheiterationcounter.Also,UBandLBdenotetheincumbentupperandlowerboundsontheobjectivefunctionvalueof 100 PAGE 101 endLagrangianRelaxation. Figure51. PseudocodeoftheLagrangianrelaxation 5{17 )and( 5{20 )withLagrangemultipliersand,respectively,whereandarematriceswithsizesNNandNK,respectively.TherelaxationyieldsthefollowingLagrangianproblem(ERSFLCPLR): minNXi=1NXj=1(tnidij+ij+niXkjk)xij+NXj=1KXk=1(fj+cjkXiijjkjk+jkMj)yjkNXj=1KXk=1jkMj 101 PAGE 102 NotethattheLagrangianproblemcanbeseparatedintosubproblemsLXandLY,wheresubproblemLXcontainsvariablesxij,andthesubproblemLYcontainsvariablesyjkasfollows: (LX)minNXi=1NXj=1~dijxijsubjecttoNXj=1aijxij=18ixij2f0;1g8i;j(LY)minNXj=1KXk=1~cjkyjksubjecttoNXj=1KXk=1yjkpKXk=1yjk18jyjk2f0;1g8j;k Givenasetofand,bothsubproblemsLXandLYareeasytosolve.ForsubproblemLX,foreachi2Nthedecisionvariablexijissettooneif~dij~dijforallj2N.Similarly,subproblemLYcanbesolvedaccordingtoeachvariableyjk's 102 PAGE 103 52 depictsthepseudocodeoftheheuristic. Theheuristicstartsbyresettingtheinfeasiblesolution(X,Y)accordingtothestrategyselectedrandomlyfromfollowing: 1.Foreachi,ifxij>0,thenPNk=1yjk>0;and 2.IfPNk=1yjk=0,thenxij=0foralli2N. Therststrategyresetsfacilityvariables,yjk,basedondemandallocation,xij,determinedfromsubproblemLX.Thefacilitiesareorderedaccordingtotheirpatientarrivalrates,i.e.,j=Pinixij,innonincreasingordern,andthentherstpfacilitiesaresettoopenatcapacitylevelone,i.e.,yj1=1.Thesecondstrategyresetsdemandallocationvariables,xij,accordingtosolutionyjkformsubproblemLY.Thatis,ifafacilityjisnotopened,thenthedemandnodesallocatedtofacilityjareresettonotcovered,i.e.,thecorrespondingvariablexij'sareresetto0andreallocatethemtootherfacilitiesusingthenextprocedure. Line 2 resetsthecapacitylevelkoftheopenedfacilitiessuchthatthefacilitiesisopenedatappropriatecapacitylevelk,i.e.,Pinixijjk.Foranopenedfacilityj,ifthedemandallocatedexceedsitslargestcapacity,i.e.,Pinixij>jK,thenanallocateddemandnodeiisselectedrandomlyanditsxijisresetto0,untilPinixijjkholds. Next,thewhileloopinline 3 inFigure 52 ensuresthatalldemandnodesarecoveredandthenumberoffacilitiesopenedislessthanp.ThepseudocodeofcoveringalldemandnodesispresentedinFigure 53 .Werandomlyselectanodeuwhichhasnotcoveredby 103 PAGE 104 Figure52. Pseudocodeofthefeasiblesolutiongeneration anyfacility,andgenerateasetOpenFwhichcontainsthefacilitiesmeetingthefollowingcriteria: 1.Thefacilityjisopened,i.e.,Pkyjk>0; 2.Thefacilityjcancoveredthedemandnodeu,i.e.,auj=1;and 3.Afterallocatingnodeutothefacilityj,thesumofarrivalratesoftheallocateddemandnodedoesnotexceedfacilityj'smaximumcapacity,i.e.,Pinixij+nujK. IfthesetOpenFisempty,thenasetNotOpenFisgenerated.ThesetNotOpenFincludesthefacilitieswhicharenotopened,i.e.,Pkyjk=0,andthenodeuiswithintheircoveragerange,i.e.,auj=1forallj2N.Then,intheline 8 inFigure 53 weopenafacilityj2NotOpenFtocovernodeu.Therearemanystrategiesthatwecanapplytochoosewhichfacilitytoopen,suchas,openthefacilityj2NotOpenFwhichistheclosestonetonodeu,theonewiththelargestcapacity,ortheonewiththesmallestxcost.Hereweapplythethreestrategiesandndthecorrespondingfacilitiesforeachofthem.Ifthesestrategiesyielddierentfacilityoptions,werandomlyselectoneofthemtoopen.Oncealldemandnodesarecoveredbytheopenedfacilities,thecapacitylevelofeachopenedfacilityisresettoappropriatelevelk2K(line 14 inFigure 53 )suchthatPinixijjk. Sofar,wehaveensuredthatalldemandnodesarecoveredbyfacilitieswhichareopened,andthedemandallocationdoesnotexceedthemaximumcapacityoftheopened 104 PAGE 105 Figure53. Pseudocodeofcoveringalldemandnodes facilities.Thenextthingtodoistoinspectwhetherthenumberoffacilitiesopenedisnolargerthanp.Ifthenumberoffacilitiesopenedislessthanorequaltop,thenafeasiblesolutionisgenerated.Otherwise,oneoftheopenedfacilitiesisselectedrandomlyandclosed,untilthenumberoffacilitiesopenedisnolargerthanp.Inaddition,theassociateddemandnodesareresettonotcovered,i.e.,resetxij=0,wherefacilityjischosentoclose. Figure54. Pseudocodeofclosingfacilities 105 PAGE 106 5{17 )and( 5{20 )by0xij1and0yjk1,respectively.Then,CPLEXisusedtosolvetheERSFLCPrproblem,andthedualinformationoftheconstraintsets0xij1and0yjk1areextractedtosettheinitialvaluesofLagrangianmultipliers.Letrandrbethedualvaluesassociatedwiththeconstraintsets0xij1and0yjk1,respectively.Themultipliers0and0ofERSFLCPLRareinitializedbytheequations0=rand0=r. WethenapplythemethoddescribedbyFisher(1981)toupdatethemultipliers.Ateachiterationn,thestepsizetnisobtainedbysn=Bn(^n) 106 PAGE 107 51 ,andtheotherthreeparametersaresetatlevel2. Table51. Experimentalfactorsettings ParametersLevel1Level2Level3 Experiment1p51020Experiment2K()5(8)8(5)10(4)Experiment30.5%1%2%Experiment4t($/perminute)2550100 Foreachexperimentalsetting,wegenerate30randominstances.Eachrandominstanceisobtainedbygeneratingparametersthatcorrespondtopatientarrivalrates,servicerates,xedandoperatingcostsanddistancebetweeneachnodepair.Specically,eachparameterisobtainedbyusingtheformula:u(meanvalueoftheparameter),whereuisdrawnfromthedistributionU[0:5;1:5].Themeanvaluesoftherandomparametersarelistedasfollows: 107 PAGE 108 Last,thecoveragerange,d,issetto50minutesforallinstances. Foreachinstance,theLagrangianrelaxationalgorithmdescribedinSection 5.4 isapplied.TheupperlimitofthenumberofLRiterations(nmax)issetat100,000andtheupperlimitofthenumberofconsecutiveiterationsfailtoimprovethebestknownfeasiblesolution(umax)issetat10,000.TheparameterBusedtomodiedthestepsizeisinitializedat2,anddividedby1.5ifthelowerboundisnotimprovedfor3Niterations. 55 depicttheupperandlowerbounds,respectively,obtainedfromtheLRiterationswiththedualinformationofERSFLCPr.ThedashlinesshowtheLRresultswithoutapplyingthedualinformationofERSFLCPr.ThedualinformationofERSFLCPrprovidesagoodlowerboundsolutionwhichguidestheheuristictondthebestupperboundsolutionearlierthantheguidanceofthelowerboundobtainedfromusingtheLagrangianmultipliersgeneratedfromscratch.TheconvergencegapsinFigure 55 aregivenby^~ where^and~aretheupperandlowerboundsofoptimalobjectivevalueofERSFLCPLRobtainedfromtheLRalgorithm. Table 52 toTable 55 reporttheresultsoffourexperiments.Ingeneral,LRperformsquitewellinsolvingtheERSFLCPproblemintermsoftheconvergencegapandthecomputationaleort(i.e.,CPUtime)required.Forthenetworkwith25nodes,LRtakesless2secondstoobtainthesolutionswithconvergencegapslessthan5%onaverage.Forthelargestnetworktested,i.e.,100nodes,theaverageCPUtimeisincreased,althoughitisstilllessthan60seconds.Moreover,LRdoesnotappeartoexperiencesignicantincreaseinconvergencegapduetoanincreaseinproblemsize. 108 PAGE 109 ConvergencespeedofthemodiedLR Therstexperimentshowsthatthesmallerthevalueofpis,thehighertheCPUtimerequiredtoobtainthesolution,sincesmallerpresultsinatighterconstraintonnumberoffacilitiesopened.Inparticular,thistrendcanbeobservedintheproblemswithsmallandmediumsizenetworks.Nevertheless,theproblemcanstillbesolvedwithin40secondsevenforlargesizenetworks. Table52. Experiment1:eectsofmaximumnumberoffacilitiesopened Networksize2550100Average GapTimeGapTimeGapTimeGapTimep(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.) 53.01.14.44.95.841.84.415.9103.10.93.04.33.835.53.313.6202.50.93.33.22.929.42.911.2 Average2.91.03.64.14.235.63.513.6 Inthesecondexperiment,weassumethatatmost40bedsareusedforanopenedER,sotheincrementalamountofbedspercapacitylevelarechangedwiththesettingofK.AlargeKvaluerepresentsthatcapacityisincreasedatsmallscale,i.e.,smallbatchsize,andincreasesthenumberofvariablesoftheunderlyingformulationthetheERSFLCPproblem.Asaresult,theCPUtimeisincreasedasKincreases,while 109 PAGE 110 Table53. Experiment2:eectsofcapacitysetting Networksize2550100Average GapTimeGapTimeGapTimeGapTimeK(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.) 5(8)0.70.82.24.23.631.42.112.18(5)3.21.12.94.13.934.03.313.110(4)3.61.13.34.53.935.93.613.8 Average2.51.02.84.33.833.83.013.0 ThethirdexperimentinvestigatestheimpactofthediversionprobabilityontheperformanceoftheproposedLRapproach.Giventhecapacitylevelk,thesmallerincursthesmallervalueofthemaximumarrivalratesthatafacilityisabletoserve,i.e.,jkinconstraintset( 5{20 ).Thatis,theconstraintset( 5{20 )becomestighterasdecreases.AsshowninTable 54 ,theCPUtimeslightlyincreasesasdecreasesfrom2.0%to1.0%.However,whenthevalueofisfurtherreducedto0.5%,therequiredCPUtimedecreases,whichisdierentfromthetrendthatweobservebefore.Theconvergencegapdoesnotappeartobeaectedbysignicantly;inaverage,thegapsare3.6%and2.9%when'sare0.5%and2.0%,respectivley. Table54. Experiment3:eectsofdiversionprobability Networksize2550100Average GapTimeGapTimeGapTimeGapTime(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.) 0.5%3.70.93.34.13.834.93.613.31.0%3.11.03.04.33.956.63.320.62.0%2.40.92.73.63.548.32.917.6 Average3.10.93.04.03.746.63.317.2 Thelastexperimentexplorestheeectsofpatients'valueoftimetontheperformanceoftheproposedLRapproach.Table 55 showsthatthehigherthepatients'timevalueis,thesmallertheconvergencegapis.WealsoobservethattheCPUtimeisnotsignicantlyaectedbythechangeoft. 110 PAGE 111 Experiment4:eectsoftimevalue Networksize2550100Average GapTimeGapTimeGapTimeGapTimet(%)(sec.)(%)(sec.)(%)(sec.)(%)(sec.) 254.91.14.95.85.645.85.117.6503.01.12.94.73.849.03.218.21001.81.11.75.42.750.82.119.1 Average3.21.13.25.34.048.53.518.3 Toillustratetheperformanceoftheheuristicdevelopedforobtainingtheupperbound(feasible)solutionateachiteration,weuseCPLEXtosolvetheERSFLCPproblem.Theparameters,p,K,andtaresettotheirrespectivevaluesforlevel2inTable 51 .Foreachtestinstance,CPLEXisstoppediftherelativestoppingtoleranceof0.01%issatised,orCPUtimeof3,600secondsisused.Table 56 summarizestheresults,wheretherelativeerrorofeachinstanceisgivenbyObjofCPLEXUpperboundofLR ObjofCPLEX100%: 56 showsthatthedevelopedheuristicisveryeective.Forsmallandmediansizeproblems,theupperboundsobtainedfromtheheuristicareveryclosetotheoptimalobjectivevalues;theaverageandmaximumrelativeerrorsarelessthan0.3%and1.5%,respectively.Fortheproblemwithlargestsize,CPLEXisnotabletondtheoptimalsolutionwithin3,600seconds.Therefore,weusethebestresultsthatCPLEXcanndwithin3,600secondstocomparewiththeresultsobtainedbyheuristic.Thecomparisonfurtherensurestheeectivenessoftheheuristicdevelopedhere.Theheuristicnotonlyspendsmuchlesstime(36secondsinaverage)thanCPLEX,butalsondsolutionswithbetterqualityin50%oftheinstances. 111 PAGE 112 Performanceoftheheuristic CPLEXHeuristicErr.CPLEXHeuristicErr.CPLEXHeuristicErr.Instance(sec.)(sec.)(%)(sec.)(sec.)(%)(sec.)(sec.)(%) 10.40.00.831.83.30.1>360026.60.420.70.01.484.34.50.1>360059.71.330.30.01.144.93.50.0>360026.00.440.50.01.1109.94.70.2>360039.70.250.80.00.882.45.10.5>360031.90.260.70.00.847.97.50.9>360033.30.170.20.00.9115.03.50.1>360029.40.580.20.00.818.74.10.0>360030.50.190.90.01.027.73.50.2>360051.40.4100.20.00.932.64.60.2>360027.80.7110.80.11.1130.95.10.5>360065.60.1120.30.00.965.25.30.2>360030.10.3130.60.00.965.53.00.1>360022.81.8140.30.00.939.94.80.6>360039.50.4150.60.00.925.63.40.3>360025.90.3160.50.00.9456.43.70.3>360055.30.8170.60.01.047.94.30.6>360034.80.4180.80.01.634.73.20.1>360041.50.2190.70.00.847.33.50.3>360019.50.2200.80.00.8119.83.81.2>360024.40.2210.40.00.853.53.80.1>360025.70.2220.40.00.846.73.40.0>360025.41.2230.20.00.817.47.50.5>360033.70.3240.30.00.862.83.30.1>360049.61.1250.50.00.9502.13.50.2>360028.10.5260.80.00.953.63.60.3>360023.61.4270.60.01.027.43.70.4>360019.90.8280.60.00.9146.64.80.3>360037.10.2290.20.00.939.95.81.2>360060.21.2300.10.00.9176.94.10.0>360046.20.2 Min.0.10.00.817.43.00.0>360019.51.8Avg.0.50.00.991.84.30.3>360035.50.3Max.0.90.11.6502.17.51.2>360065.60.7 PAGE 113 Animmediateextensionofourmodelistoincludetheclosestassignmentconstraintswhichensurethateachdemandpointisallocatedtotheclosestopenfacility.Someotherpracticallyrelevantvariationsareconcernedwiththealternativeobjectivefunctions,suchasprotmaximization,andtravelandservicetimesminimization.AnothercloselyrelatedproblemisthatgivensomeERservicefacilitiesareopenedalreadyandBnewbedsareconsideredtobeaddedatexistingfacilitiesornewlyopenedfacilities.Wenotethatthiscanbedenedbyaddingtheconstraintsoncapacityavailable,distinguishingthesetofexistingandnewERlocations,andreplacingthexedcosttocapacityexpansioncostofexistingfacilities. 113 PAGE 114 Inthiswork,wehavepresentedtheintegrateduseofoptimizationandqueueingtheorytodeterminetheoptimalcapacityplanforhealthcaresystems. Chapter 2 detailedtheaggregatehospitalbedcapacityplanning(AHBCP)problemandanetworkowapproachtospecifytheoptimalbedcapacityplanningdecisionsthroughoutaniteplanninghorizonforhospitals.Inthischapter,ahospitalwasmodeledasaG=G=cqueueingsystemwithasinglebedtypeandasinglepatientclass.Wedemonstratedthatforrealisticsizedcapacityplanningproblems,ournetworkformulationisnotcomputationallyintensive,andallowsustoobtainoptimalbedcapacityplansquickly. Chapter 3 introducedthehealthcareteamcapacityplanning(HCTCP)problem,inwhichtheunderlyingqueueingsystemwasmorecomplexthantheoneusedforAHBCP.Inparticular,weconsideredaqueueingsystemwheretherearetwoclassesofpatientsandtwotypesofcareteams,whereserviceratesarepatientclassdependentandonetypeofcareteamcansubstitutefortheother.Wedevelopedqueueingmodelsforbothpreemptiveandnonpreemptivecases,anddevelopedapproximationprocedurestoestimatetheaveragetimethateachpatientclassspendsinthesystem.Theresultsfromapproximationmethodwerethenincorporatedintotheoptimizationmodeltodeterminetheminimalcostcapacityplanofhealthcareteamsthroughoutaniteplanninghorizon.Ourcomputationalstudyshowedthatourapproximationapproachprovidessucientlyaccurateresultsthatcanbeusedinpracticetomakelongtermhealthcareteamservicecapacityplanningdecisions. AftertheaggregatebedcapacitywasspeciedinChapter 2 ,wedevelopedthehospitalbedallocation(HBA)modeltoobtainthebalancedbedallocationamongdierentmedicalcareunits,whichweremodeledasM=M=c=cqueueingsystems.Toecientlysolvetheproblem,weproposedthreesolutionapproachesincluding 114 PAGE 115 Chapter 5 developedaemergencyroomservicesfacilitylocationandcapacityplanning(ERSFLCP)model,whereeachemergencyroomservicefacilityisviewedasaM=M=c=cqueueingsystem.Themodelwasdesignedtosimultaneouslydeterminethenumberoffacilitiesopenedandtheirrespectivelocationsaswellasthecapacitylevelsofthefacilities(captureintermsofnumberofbeds)sothattheprobabilitythatthefacilityisfullandincomingpatientshavetobediverted(i.e.,diversionprobability)isnotlargerthanaparticularthreshold.ALagrangianrelaxationapproachwasproposedtoobtainfacilitylocationsandcapacityplan.TheexperimentalresultsillustratedthattheLagrangianrelaxationalgorithmisveryecientinsolvingtheproblem,andthedevelopedheuristicprovidessolutionswithgoodquality. Inourworktodate,weprimarilyfocusedonstrategiclevel,longtermplanningdecisionswherewemadesomesimplifyingassumptionsastothecapabilitiesoftheresourcesandthearrivalratesandlengthsofstayforthepatients.Specically,weassumedthattheavailablebedsinaservice(inChapters 3 4 ,and 5 )orthehospital(inChapter 2 )areidentical.Similarly,weassumedthatthearrivalratesandlengthofstayforthepatientsareidentical(inChapters 2 and 5 ).Butwealsoconsideredthecasewherepatientscanbegroupedintotwoclasses(eachofwhichcorrespondstoanacuitylevel)orintomultipleclasses(eachofwhichcorrespondstoaparticularspeciality)tomodelarrivalratesandlengthsofstay.Aswemainlyfocusedonstrategicleveldecisionmaking,theseassumptionsarejustiable.However,formoredetailedplanningthereisaneedtotakeotherrealisticconsiderationsintoaccount. Therearetypicallymultipletypesofbeds(e.g.,adultintensivecarebeds,pediatricintensivecarebeds,burnintensivecarebeds,medical/surgicalbeds),i.e.,multiple 115 PAGE 116 Inourwork,wehaveprimarilyfocusedontheobjectiveofminimizingtotalcosts(inChapters 2 3 ,and 5 )andbalancingworkloadacrossunits(inChapter 4 ).Nowadays,hospitalsarefocusingonrevenuemanagementpracticestoimprovetheirnancialsituation.Thismodernrevenuemanagementculturerequireshospitaladministratorstofocusonmaximizingprotratherthanonminimizingoperatingcosts.Therefore,futureworkshouldexaminetherevenueaspectsofhospitaloperationsandfocusonprotmaximizationtypeobjectives.However,thisshiftinemphasisfromcostminimizationtoprotmaximizationshouldnotignorethequalityaspectsassociatedwiththedeliveryofhealthcareservices.Inourresearch,wefocusedontimeliness(e.g.,averageservicetime,averagewaitingtime)andaccess(e.g.,diversionprobability)toquantifyservicequality.Futureresearchintheareashouldconcentrateondevelopingothermetricstomodelotheraspectsofservicequality,suchaspatientsafetyandserviceeectiveness. 116 PAGE 117 [1] AgencyforHealthcareResearchandQuality.Lasttimeaccessed:November2007,http://www.ahrq.gov/research/havbed/definitions.htm. [2] R.K.Ahuja,T.L.Magnanti,J.B.Orlin,NetworkFlows:Theory,Algorithms,andApplications,PrenticeHall,NewJersey,1993. [3] L.A.Aiken,S.P.Clarke,D.M.Sloaneetal.,Nursesreportsonhospitalcareinvecountries,HealthAairs20(2001)4353. [4] L.A.Aiken,S.P.Clarke,D.M.Sloaneetal.,Hospitalnursestangandpatientmortality,nurseburnout,andjobdissatisfaction,JAMA288(2002)1987{1993. [5] E.Akcal,M.J.C^ote,C.ILin,Anetworkowapproachtooptimizinghospitalbedcapacitydecisions,HealthCareManagementScience9(2006)391{404. [6] R.Akkerman,M.Knip,Reallocationofbedstoreducewaitingtimeforcardiacsurgery,HealthCareManagementScience7(2004)119{126. [7] AmericanHospitalAssociation,HospitalStatistics2005Edition,HealthForum,Chicago,2005. [8] P.A.Andersson,E.Varde,F.Diderichsen,ModellingofresourceallocationtohealthcareauthoritiesinStockholmCounty,HealthCareManagementScience3(2000)141{149. [9] R.Batta,O.Berman,Q.Wang,Stangandswitchingcostsinaservicecenterwithexibleservers,EuropeanJournalofOperationalResearch177(2007)924{938. [10] G.J.Bazzoli,L.R.Brewster,G.Liu,S.Kuo,DoesU.S.hospitalcapacityneedtobeexpanded?HealthAairs22(2003)4054. [11] D.Bellandi,Runningatcapacity,ModernHealthcare110(1999)112113. [12] P.Beraldi,M.E.Bruni,D.Conforti,Designingrobustemergencymedicalserviceviastochasticprogramming,EuropeanJournalofOperationalResearch158(2004)183{193. [13] O.Berman,D.Krass,J.Wang,LocatingServiceFacilitiestoReduceLossDemand,IIETransactions38(2006)933{946. [14] O.Berman,R.Huang,S.Kim,S.,M.Menezes,Locatingcapacitatedfacilitiestomaximizecaptureddemand,IIETransactions39(2007)1015{1029. [15] D.M.Berwick,Wecancutcostsandimprovecareatthesametime,MedicalEconomics180(1996)185187. [16] G.R.Bitran,D.Tirupati,Capacityplanninginmanufacturingnetworkswithdiscreteoptions,AnnalsofOperationsResearch17(1989)119{136. 117 PAGE 118 G.R.Bitran,D.Tirupati,Tradeocurves,targetingandbalancinginmanufacturingqueueingnetworks,OperationsResearch37(1989)547{564. [18] J.T.Blake,M.W.Carter,Agoalprogrammingapproachtostrategicresourceallocationinacutecarehospitals,EuropeanJournalofOperationalResearch140(2002)541{561. [19] J.P.C.Blanc,Performanceevaluationofpollingsystemsbymeansofthepowerseriesalgorithm,AnnalsofOperationsResearch35(1992)155{186. [20] J.P.C.Blanc,Performanceanalysisandoptimizationwiththepowerseriesalgorithm,inL.Donatiello,R.Nelson(eds),PerformanceanalysisandoptimizationwithpowerseriesalgorithmSpringerVerlag,Berlin,53{80,1993. [21] B.Boey,D.Yates,R.D.Galv~ao,AnalgorithmtolocateperinatalfacilitiesinthemunicipalityofRiodeJaneiro,JournaloftheOperationalResearchSociety54(2003)21{31. [22] J.Bowers,B.Lyons,G.Mould,T.Symonds,Modellingoutpatientcapacityforadiagnosisandtreatmentcentre,HealthCareManagementScience8(2005)205{211. [23] J.Bowers,G.Mould,Managinguncertaintyinorthopaedictraumatheatres,EuropeanJournalofOperationalResearch154(2004)599{608. [24] J.Bowers,G.Mould,Ambulatorycareandorthopaediccapacityplanning,HealthCareManagementScience8(2005)41{47. [25] M.L.Brandeau,F.Sainfort,W.P.Pierskalla,OperationsResearchandHealthCare:AHandbookofMethodsandApplications,Springer,Boston,2004. [26] K.M.Bretthauer,M.J.C^ote,Amodelforplanningresourcerequirementsinhealthcareorganizations,DecisionSciences29(1998)243{270. [27] M.J.Brusco,M.J.Showalter,Constrainednursestaanalysis,OMEGA:InternationalJournalofManagementScience21(1993)175{186. [28] E.K.Burke,P.DeCausmaecker,G.VandenBerghe,H.VanLandeghem,Thestateoftheartofnurserostering,JournalofScheduling7(2004)441{499. [29] F.Cerne,J.Montague,Capacitycrisis,HospitalsandHealthNetworks68(19)(1994)30{36. [30] J.K.Cochran,A.Bharti,Stochasticbedbalancingofanobstetricshospital,HealthCareManagementScience9(2006)31{45. [31] R.C.CoileJr.,Futurescan2002:aforecastofhealthcaretrends20022006,HealthAdministration,Chicago,2002. [32] M.J.C^ote,S.L.Tucker,Fourmethodologiestoimprovehealthcaredemandforecasting,HealthcareFinancialManagement55(2001)5458. 118 PAGE 119 R.Davies,Simulationforplanningservicesforpatientswithcoronaryarterydisease,EuropeanJournalofOperationalResearch72(1994)323{332. [34] K.Davis,S.C.Schoenbaum,A.M.Audet,A2020VisionofPatientCenteredPrimaryCare,JournalofGeneralInternalMedicine20(2005)953{957. [35] L.Delesie,A.Kastelein,F.Merode,J.M.H.Vissers,Managinghealthcareunderresourceconstraints,EuropeanJournalofOperationalResearch105(1998)247{247. [36] T.A.Feo,M.G.C.Resende,Aprobabilisticheuristicforacomputationallydicultsetcoveringproblem,OperationsResearchLetters8(1989)6771. [37] M.L.Fisher,TheLagrangianrelaxationmethodforsolvingintegerprogrammingproblems,ManagementScience27(1981)1{18. [38] S.Flessa,Whereeciencysaveslives:Alinearprogrammefortheoptimalallocationofhealthcareresourcesindevelopingcountries,HealthCareManagementScience3(2000)249{267. [39] R.D.Galv~ao,L.G.A.Espejo,B.Boey,AhierarchicalmodelforthelocationofperinatalfacilitiesinthemunicipalityofRiodeJaneiro,EuropeanJournalofOperationalResearch138(2002)495{517. [40] R.D.Galv~ao,L.G.A.Espejo,B.Boey,D.Yates,Loadbalancingandcapacityconstraintsinahierarchicallocationmodel,EuropeanJournalofOperationalResearch172(2006)631{646. [41] D.E.Goldberg,GeneticAlgorithmsinSearch,OptimizationandMachineLearning,AddisonWesley,Wokingham,England,1989. [42] S.M.Goldstein,P.T.Ward,G.K.Leong,T.W.Butler,Theeectoflocation,strategy,andoperationstechnologyonhospitalperformance,JournalofOperationsManagement20(2002)63{75. [43] F.Gorunescu,S.I.McClean,P.H.Millard,Usingaqueueingmodeltohelpplanbedallocationinadepartmentofgeriatricmedicine,HealthCareManagementScience5(2002)307{312. [44] L.Green,Aqueuingsystemwithgeneraluseandlimiteduseservers,OperationsResearch33(1985)168{182. [45] L.V.Green,Howmanyhospitalbeds?Inquiry38(2002/2003)400412. [46] L.V.Green,V.Nguyen,Strategiesforcuttinghospitalbeds:theimpactonpatientservice,HealthServicesResearch36(2001)421{442. [47] S.Groothuis,A.Hasman,P.E.J.vanPoletal.,Predictingcapacitiesrequiredincardiologyunitsforheartfailurepatientsviasimulation,ComputermethodsandProgramsinBiomedicine74(2)(2004)129{141. 119 PAGE 120 D.Gross,C.M.Harris,FundamentalsofQueueingTheory(3nded.),JohnWileyandSons,NewYork,1998. [49] K.Grumbach,T.Bodenheimer,Canhealthcareteamsimproveprimarycarepractice?JAMA:thejournaloftheAmericanMedicalAssociation291(2004)1246{1251. [50] C.Hayward,Whatarewegoingtodowithallourexcesscapacity?HealthCareStrategicManagement16(1998)20{23. [51] P.R.Harper,Aframeworkforoperationalmodellingofhospitalresources,HealthCareManagementScience5(2002)165{173. [52] G.W.Harrison,A.Shafer,M.Mackay,ModellingVariabilityinHospitalBedOccupancy,HealthCareManagementScience8(2005)325{334. [53] J.H.Holland,AdaptationinNaturalandArticialSystems,UniversityofMichiganPress,AnnArbor,1975. [54] G.Hooghiemstra,M.Keane,S.V.D.Ree,Powerseriesforstationarydistributionsofcoupledprocessormodels,SIAMJournalonAppliedMathematics48(1988)1159{1166. [55] InstituteofMedicine,CrossingtheQualityChasm,NationalAcademyPress,Washington,D.C,2001. [56] H.Jia,F.Ordonez,M.M.Dessouky,Amodelingframeworkforfacilitylocationofmedicalservicesforlargescaleemergencies,IIETransactions39(2007)41{55. [57] A.M.Johnson,Capacityplanningforthefuture,JournalofHealthCareFinance24(1997)7275. [58] J.B.Jun,S.H.Jacobson,J.R.Swisher,Applicationofdiscreteeventsimulationinhealthcareclinics:Asurvey,JournaloftheOperationalResearchSociety50(2)(1999)109{123. [59] E.P.C.Kao,C.Lin,TheM/M/1queuewithrandomlyvaryingarrivalandservicerates:aphasesubstitutionsolution,ManagementScience35(1989)561{570. [60] E.P.C.Kao,K.S.Narayanan,Computingsteadystateprobabilitiesofanonpreemptiveprioritymultiserverqueue,ORSAJournalonComputing2(1990)211{218. [61] E.P.C.Kao,M.Queyranne,Budgetingcostsofnursinginahospital,ManagementScience31(1985)608{621. [62] E.P.C.Kao,G.G.Tung,Aggregatenursingrequirementplanninginapublichealthcaredeliverysystem,SocioEconomicsPlanningScience15(1981)119{127. 120 PAGE 121 E.P.C.Kao,G.G.Tung,Bedallocationinapublichealthcaredeliverysystem,ManagementScience27(1981)507{520. [64] E.P.C.Kao,S.D.Wilson,Analysisofnonpreemptivepriorityqueueswithmultipleserversandtwopriorityclasses,EuropeanJournalofOperationalResearch118(1999)181{193. [65] E.H.Kaplan,M.Johri,Treatmentondemand:anoperationalmodel,HealthCareManagementScience3(2000)171{183. [66] S.Kavanagh,J.Cowan,Reducingriskinhealthcareteams:anoverview,ClinicalGovernance9(2004)200204. [67] S.Kim,I.Horowitz,K.K.Young,T.A.Buckley,Analysisofcapacitymanagementoftheintensivecareunitinahospital,EuropeanJournalofOperationalResearch115(1999)36{46. [68] S.Kim,I.Horowitz,K.K.Young,T.A.Buckley,Flexiblebedallocationandperformanceintheintensivecareunit,JournalofOperationsManagement18(2000)427{443. [69] L.Kleinrock,Queueingsystems:TheoryVol.1,Wiley,NewYork,1975. [70] N.Koizumi,E.Kuno,T.E.Smith,Modelingpatientowsusingaqueuingnetworkwithblocking,HealthCareManagementScience8(2005)49{60. [71] G.Koole,OntheuseofthepowerseriesalgorithmforgeneralMarkovprocesses,withanapplicationtoaPetrinet,INFORMSJournalonComputing9(1997)51{56. [72] D.H.Kropp,R.C.Carlson,Recursivemodelingofambulatoryhealthcaresettings,JournalofMedicalSystems1(1977)123{135. [73] R.J.Kusters,P.M.A.Groot,Modellingresourceavailabilityingeneralhospitalsdesignandimplementationofadecisionsupportmodel,EuropeanJournalofOperationalResearch88(1996)428{445. [74] G.Latouche,V.Ramaswami,Alogarithmicreductionalgorithmforquasibirthanddeathprocesses,JournalofAppliedProbability30(1993)650{674. [75] G.Latouche,V.Ramaswami,IntroductiontoMatrixAnalyticMethods,SIAM,Philadelphia,PA,1999. [76] L.X.Li,W.C.Benton,Performancemeasurementcriteriainhealthcareorganizations:Reviewandfutureresearchdirections,EuropeanJournalofOperationalResearch93(1996)449{468. [77] L.X.Li,W.C.Benton,G.K.Leong,Theimpactofstrategicoperationsmanagementdecisionsoncommunityhospitalperformance,JournalofOperationsManagement20(2002)389{408. 121 PAGE 122 E.Litvak,P.I.Buerhaus,F.Davido,F.etal.,Managingunnecessaryvariabilityinpatientdemandtoreducenursingstressandimprovepatientsafety,JointCommissionJournalonQualityandPatientSafety31(2005)330338. [79] M.Mackay,Practicalexperiencewithbedoccupancymanagementandplanningsystems:anAustralianview,HealthCareManagementScience4(2001)47{56. [80] M.Mackay,M.Lee,ChoiceofModelsfortheAnalysisandForecastingofHospitalBeds,HealthCareManagementScience8(2005)221{230. [81] V.Marianov,C.ReVelle,Thequeueingmaximalavailabilitylocationproblem:Amodelforthesitingofemergencyvehicles,EuropeanJournalofOperationalResearch93(1996)110{120. [82] A.H.Marshall,S.I.McClean,C.M.Shapcott,P.H.Millard,Modellingpatientdurationofstaytofacilitateresourcemanagementofgeriatrichospitals,HealthCareManagementScience5(2002)313{319. [83] A.H.Marshall,C.Vasilakis,E.ElDarzi,Lengthofstaybasedpatientowmodels:recentdevelopmentsandfuturedirections,HealthCareManagementScience8(2005)213{220. [84] B.J.Masterson,T.G.Mihara,G.Miller,S.C.Randolph,M.E.Forkner,A.L.Crouter,Usingmodelsanddatatosupportoptimizationofthemilitaryhealthsystem:Acasestudyinanintensivecareunit,HealthCareManagementScience7(2004)217{224. [85] W.E.McAleer,I.A.Naqvi,Therelocationofambulancestations:Asuccessfulcasestudy,EuropeanJournalofOperationalResearch75(1994)582{588. [86] L.W.Morton,N.Bills,D.Kay,Boostinglocaleconomiesthroughhealthcareeconomicdevelopment,Lasttimeaccessed:August2005,http://www.cardi.cornell.edu/economic development/community economic renewal/000274.phpCommunityandRuralDevelopmentInstitute,CornellUniversity. [87] M.F.Neuts,MatrixGeometricSolutionsinStochasticModels,JohnsHopkinsUniversityPress,Maryland,1981. [88] J.M.Nguyen,P.Six,T.Chaussalet,D.Antonioli,P.Lombrail,P.LeBeux,AnobjectivemethodforbedcapacityplanninginahospitaldepartmentAcomparisonwithtargetratiomethods,Methodsofinformationinmedicine46(4)(2007)399{405. [89] W.P.Pierskalla,Healthcaredelivery,PresentedattheNationalScienceFoundationWorkshoponEngineeringtheServiceSector,Atlanta,2001. 122 PAGE 123 W.P.Pierskalla,D.Brailer,Applicationsofoperationsresearchinhealthcaredelivery,inBeyondtheprotmotive:publicsectorapplicationsandmethodology,HandbooksinOR&MS,S.Pollock,A.Barnett,M.Rothkopf(eds),Vol6,NorthHolland,NewYork,1994. [91] W.P.Pierskalla,D.Wilson,Reviewofoperationsresearchimprovementsinpatientcaredeliverysystems,Workingpaper,UniversityofPennsylvania,Philadelphia,1989. [92] S.Rahman,D.K.Smith,Useoflocationallocationmodelsinhealthservicedevelopmentplanningindevelopingnations,EuropeanJournalofOperationalResearch123(2000)437{452. [93] J.F.Repede,J.J.Bernardo,DevelopingandvalidatingadecisionsupportsystemforlocatingemergencymedicalvehiclesinLouisville,Kentucky,EuropeanJournalofOperationalResearch75(1994)567{581. [94] J.C.Ridge,S.K.Jones,M.S.Nielsen,A.K.Shahani,Capacityplanningforintensivecareunits,EuropeanJournalofOperationalResearch105(1998)346{355. [95] S.M.Ryan,Capacityexpansionforrandomexponentialdemandgrowthwithleadtimes,ManagementScience50(2004)740{748. [96] F.Sainfort,WhereisOR/MSinthepresentcrisesinhealthcaredelivery?PresentedattheInstituteforOperationsResearchandtheManagementSciencesAnnualMeeting,Miami,2001. [97] D.P.Schneider,K.E.Kilpatrick,AnOptimumManpowerUtilizationModelforHealthMaintenanceOrganizations,OperationsResearch23(1975)869{889. [98] L.Shi,D.A.Singh,EssentialsoftheUSHealthCareSystem,JonesandBartlettPublishers,Maryland,2005. [99] R.A.Shumsky,Approximationandanalysisofaqueueingsystemwithexibleandspecializedservers,ORSpectrum,SpecialIssueonCallCenterManagement26(3)(2004)307{330. [100] D.Sinreich,Y.Marmor,Emergencydepartmentoperations:thebasisfordevelopingasimulationtool,IIETransactions37(3)(2005)233{245. [101] V.L.SmithDaniels,S.B.Schweikhart,D.E.SmithDaniels,Capacitymanagementinhealthcareservices:reviewandfutureresearchdirections,DecisionScience19(1988)899918 [102] C.Smith,C.Cowan,A.Sensenig,A.Catlin,HealthSpendingGrowthSlowsin2004,HealthAairs25(2006)186{196. [103] L.Snyder,Facilitylocationunderuncertainty:areview,IIETransactions38(2006)537{554. 123 PAGE 124 L.Snyder,M.Daskin,ReliabilityModelsforFacilityLocation:TheExpectedFailureCostCase,TransportationScience39(3)(2005)400{416. [105] D.A.Stanford,W.K.Grassmann,Thebilingualserversystem:aqueueingmodelfeaturingfullyandpartiallyqualiedservers,INFOR31(4)(1993)261{277. [106] J.Thorson,Lasttimeaccessed:August2006,http://sepwww.stanford.edu/oldreports/sep20/20 11 abs.html. [107] UnitedStatesDepartmentofLaborBureauofLaborStatistics,Lasttimeaccessed:August2006,http://www.bls.gov/oco/cg/cgs035.htm#emply. [108] M.Utley,S.Gallivan,K.Davis,P.Daniel,P.Reeves,J.Worrall,Estimatingbedrequirementsforanintermediatecarefacility,EuropeanJournalofOperationalResearch150(2003)92{100. [109] A.H.vanZon,G.J.Kommer,Patientowsandoptimalhealthcareresourceallocationatthemacrolevel:adynamiclinearprogrammingapproach,HealthCareManagementScience2(1999)87{96. [110] J.M.H.Vissers,Patientowbasedallocationofinpatientresources:Acasestudy,EuropeanJournalofOperationalResearch105(1998)356{370. [111] T.D.Vries,R.E.Beekman,Applyingsimpledynamicmodellingfordecisionsupportinplanningregionalhealthcare,EuropeanJournalofOperationalResearch105(1998)277{284. [112] S.Yan,ApproximatingReducedCostsunderDegeneracyinaNetworkFlowProblemwithSideConstraints,Networks27(1996)267{278. 124 PAGE 125 ChinILinwasborninTaipei,Taiwan.ShereceivedherB.S.andM.SincivilengineeringfromtheNationalCentralUniversityinTaiwanin1994and1996,respectively.From1997to2002,sheworkedforChinaAirlines,andhermajortasksincludeddemandforecast,marketanalysis,routeanalysis,andeetplanning.ShepursuedhermasteranddoctoraldegreesintheDepartmentofIndustrialandSystemsEngineeringattheUniversityofFloridasince2002.ChinI'smainresearchinterestisOperationsResearch,andtopicsofspecialinterestarehealthcaremanagementandairlineight/crewscheduling.Thusfar,herresearchhasfocusedoncapacitymanagementinhealthcaredelivery. 125 